The following is a list of WordPress plugins that I find very useful.

Check out this list: wpsite care – Best WordPress plugins

I will update this post on occasion as I remember to add additional plugins I like/use.

Backup

Duplicator (Backups and clones a site). http://wordpress.org/plugins/duplicator/
It packs up the site into a zip file and creates an installer.php. It is easy to download both and install the site another place.

For instance. I plan on creating a major change to the site but do not want to do so on the live (online) site, so I use Duplicator to backup and download and install a local site through MAMP (on my Mac). I make the changes locally and add an index.html that says maintenance. Delete the online site and then upload the new site, go to the location of the installer.php and reinstall the site. It should not take too long and should be done at a period of time when there is little traffic on the site.

I added the Duplicator first since I have been helping to test out this plugin for the last year or so and find it really useful in backing up my sites. My Duplicator section.

Check out this article: pro plugin directory – best backup WordPress plugins.

Auto updating of themes, plugins etc

Easy Updates Manager
wordpress.org/plugins/stops-core-theme-and-plugin-updates

The Automatic updater by Pento.
https://wordpress.org/plugins/automatic-updater/

Update control by George and Chip.
https://wordpress.org/plugins/update-control/

Simple CSS edits

If your using a theme and just want to make some simple CSS edits without having to make a child theme then Simple Custom CSS is a plugin to try out as it will add the custom CSS that will not be overwritten when updating the theme.

Encodes open e-mail

Email Address Encoder
This plugin hides the e-mail addresses from software/robots that can pick out e-mail addresses to used to send spam. wordpress.org/plugins/email-address-encoder

If you plan on showing e-mail addresses on your site it is a good idea to use the above plugin or perhaps the default WordPress built inn feature is good enough……

Contact Forms

Ninja Forms.
Ninja Forms is a good contact form plugin.
wordpress.org/plugins/ninja-forms

Contact Form 7 is also a good contact form to use. I also have a customization tutorial on my site.
wordpress.org/plugins/contact-form-7

Gravity Forms is a plugin I use on most of my sites. I have purchased the developer license for unlimited sites.
www.gravityforms.com

Customize contents of the Sidebar

Content Aware Sidebars
http://wordpress.org/plugins/content-aware-sidebars/
This is the best sidebar plugin that I know of and I use it a lot of various sites. (Actually I still need to change over to this plugin on a few sites).

Another really good sidebar plugin is:

Woo Sidebars.
An easy to use sidebar plugin. To create additional sidebars where one can choose where to place the sidebars.
http://wordpress.org/plugins/woosidebars/
With Woo Sidebars you can create multiple sidebars to be used in specific pages of your site.

Easier backend navigation plugins

Orbisius Quick Nav makes going from one page to another page (one post to another post) just a drop down away.
A very easy and time saving plugin!
http://wordpress.org/plugins/orbisius-quick-nav/

The Intuitive Custom Post Order plugin with the drag and drop features of posts and pages into the order that I want them in is something I use on a few sites.
http://wordpress.org/plugins/intuitive-custom-post-order/

Media Library plugin

The Enhanced Media Library plugin makes it possible to tag/categorize your images. If you are using lots of media I would suggest using this plugin.
http://wordpress.org/plugins/enhanced-media-library/

Resizing of images plugin

I have used Imsanity on a few web sites as users have uploaded images directly from their digital camera to the WordPress web site. To resize the images I have used Imsanity.
http://wordpress.org/plugins/imsanity/

Google Dashboard stats

Google Analytics Dashboard for WP
The following is a really good Google analytics Dashboard plugin I use on most of my sites.
http://wordpress.org/plugins/google-analytics-dashboard-for-wp/

At Github:
https://github.com/deconf/google-analytics-dashboard-for-wp

Google Analytics Dashboard
This plugin also has post stats on the all posts screen:
http://wordpress.org/plugins/google-analytics-dashboard/

Google Analyticator.
It shows a nice backend view of how many visitors come to the site and from where.
http://wordpress.org/plugins/google-analyticator/

To use the the above plugins one also needs to create a Google Analytics account.
http://mashable.com/2011/05/23/how-to-use-google-analytics/

SEO Plugins (Search Engine Optimization)

All in One SEO
http://wordpress.org/plugins/all-in-one-seo-pack/
OR
Yoast WordPress SEO
http://wordpress.org/plugins/wordpress-seo/

Both of these plugins are really good. Test them both and see which you like.

Custom profile image instead of using a Gravatar

Avatar Manager.
An alternative to creating a Gravatar at gravatar.com is to use the plugin Avatar Manager.
http://wordpress.org/plugins/avatar-manager/

WordPress and a other sites use Gravatars to show an image with your profile.
But you can use plugins like the Avatar Manager to instead upload an image as your profile image.

Author box plugins

I really like the WordPress author box lite plugin.
http://wordpress.org/plugins/wp-author-box-lite/

For additional features there is also a pro version.
I will add some more info on the above plugin at a later time.

Newsletter plugin

MailPoet Newsletters
A good Newsletter plugin.
http://wordpress.org/plugins/wysija-newsletters/

If you need to send out newsletters one way is to use the MailPoet Newsletters plugin.
To create newsletters inside your WordPress Backend.

Creating a child theme

Orbisius Child Theme Creator
Creates a child theme of the original theme.
http://wordpress.org/plugins/orbisius-child-theme-creator/

If you plan on making CSS customizations to your site. The best method is to create a Child Theme.
A theme developer will update his/her theme on occasion and the update can override whatever changes you have done.
To save the hassle of having to recreate the changes again it is recommended to create a Child Theme.
In the Child theme you only add changes that are different then the original/parent theme.
The Child Theme Creator plugin makes it very easy to create a copy/child of your parent theme. Which you can use to modify the CSS code and other things.

Cache plugin

WP Super Cache
This plugin helps speed up the site.
http://wordpress.org/plugins/wp-super-cache/

One can check a site at for instance: http://tools.pingdom.com/fpt/ To check the speed of the site.

Another Cache plugin to check is:
http://wordpress.org/plugins/wp-fastest-cache/

Check out:
.https://hostadvice.com/blog/8-tips-for-improving-your-site-speed/

Optimize the WordPress database

WP Optimize
Cleans and optimizes the database.
http://wordpress.org/plugins/wp-optimize/

In addition WPDBSpring Clean which cleans up unused plugin entries. As one deletes a plugin it can leave entries in your database that takes up space. It is good to remove these with http://wordpress.org/plugins/wpdbspringclean/

Security

Better WP security (has now become iThemes Security)
https://wordpress.org/plugins/better-wp-security/

http://blog.joemanna.com/plugin-review-ithemes-security/

Bruteprotect – will soon become a part of Jetpack
https://wordpress.org/plugins/bruteprotect/

Wordfence
https://wordpress.org/plugins/wordfence/

It also has a caching engine they call Falcon making WordFence have atleast two functions. One for security and one for speeding up the site.

Resource site:
http://www.itechks.com/journal/wordpress/top-10-security-plugins-for-wordpress-2014/

101 securing WordPress
http://wpsecure.net/secure-wordpress/

Antivirus
http://wordpress.org/plugins/antivirus/

Akismet takes care of spam and is included in all WordPress installations.
http://wordpress.org/plugins/akismet/

Calendar plugin

All In One Event Calendar
An easy to use Calendar – free and pro version.
http://wordpress.org/plugins/all-in-one-event-calendar/ 

Other suggestions are:
Events Manager – free and pro version.
http://wordpress.org/plugins/events-manager/ 

The Events calendar by Modern Tribe – free and pro version
https://wordpress.org/plugins/the-events-calendar/

Maintenance/under construction plugin

Ultimate Maintenance Mode
This is a very good and intuitive plugin. One of the best User interfaces I have seen for a plugin in WordPress.
Check out my review/walk through: Ultimate Maintenance Mode video tutorial and screenshots

The free version can be downloaded here:
http://wordpress.org/plugins/ultimate-maintenance-mode/

When your not ready to show your site. You can choose to use this plugin so that only the logged in user will see what the site really looks like at the present stage. Everyone else will see a maintenance page/landing page (and not the full site being build or changed). It is highly recommended if you need a landing page or a maintenance page. If your a plugin developer I will also suggest to download this plugin to check out the UI and see how options are done and how they are ordered.

Translation plugin

A translation plugin I have used on a few sites is
Transposh
http://wordpress.org/plugins/transposh-translation-filter-for-wordpress/
A frontend translator. Makes it easy to quickly translate auto or by going in yourself to adjust.

Social plugin

Floating Social Bar
http://wordpress.org/support/plugin/floating-social-bar
An easy to use social plugin.

Premium plugins meaning plugins you have to pay for. Do remember that most plugins can be used for free and most plugin developers do appreciate a donation.

Layer Slider. $15
Is a very cool Slider with a good User Interface.
http://codecanyon.net/item/layerslider-responsive-wordpress-slider-plugin-/1362246

Revolution Slider is also a very good Slider plugin. $16
http://codecanyon.net/item/slider-revolution-responsive-wordpress-plugin/2751380

Menu Management Enhancer. $8
A good way to move a group of menu items at once.
http://codecanyon.net/item/menu-management-enhancer-for-wordpress/529353

Full Width Audio Player. $16
An easy to use Audio Player. There might be some free ones out there that are easier to use right now. At the time I purchased this plugin I liked this the best. If you know of another you think is better then do let me know.
http://codecanyon.net/item/fullwidth-audio-player-wordpress-plugin/1321160

Plugins I have briefly looked at (or not) but have peaked my interest!

MaxButtons
http://wordpress.org/plugins/maxbuttons/
Makes creating buttons and adding them very easy!

Intelliwidget per page featured posts and menus
http://wordpress.org/plugins/intelliwidget-per-page-featured-posts-and-menus/
Not sure about this one…..

Easy media gallery
http://wordpress.org/plugins/easy-media-gallery/
Not sure about this one either…..

Mockup
An interesting plugin for showing mockups.
http://wordpress.org/plugins/mockup/ 

There are some categories I have not mentioned but I wish I could.

I would like to find a good gallery/image plugin with a great UI.
A columns plugin where one can actually see the columns inside the content window as perhaps similar to a table. So that I can place the content in the correct column.
A plugin to extend the ease of use in creating content.

When you log into your account at wordpress.org. Select the menu item plugins. Look above to the right. Beside your name it should say something like Favorites. In the header of each plugin you view their is a heart icon with the words Favorite next to it. Clicking the word Favorite makes the plugin Favorited by you. It becomes listed on your Favorites page. To view your Favorites page click the word My Favorites next to your name.

A site to check out:
chooseplugin.com