Here is a short list of a few plugins that we use on virtually every site. I selected the items on this short list to keep things simple and relevant to every WordPress installation regardless of size, traffic, content, etc. You’ll probably need other plugins, but these are the key ones to install first.
Wordfence is a complete anti-virus and firewall package for your WordPress website and it comes in a free and premium version. Pony up for the premium version to unlock key features. WordFence has blocked tens of thousands of hacking attempts on this blog alone and this blog isn’t all that popular! If you get any amount of traffic, you will have hacking attempts, so it’s critical to have a real time scanner. Check it out at http://www.wordfence.com/ and http://wordpress.org/plugins/wordfence/
Duplicator is the easiest way I’ve found to back up and restore sites. With a few mouse clicks, you have a simple zip archive of your entire site (database and files) and can easily restore a corrupted site or move to a new host. This plugin is still in beta (as of 4/14/2014), but it’s solid. Check it out at http://lifeinthegrid.com/labs/duplicator/ and http://wordpress.org/plugins/duplicator/. This one is free but accepts donations.
Stop spam by having all of your comments run through the Akismet filter. Simple and lightweight to use. Check it out https://akismet.com/ and https://wordpress.org/plugins/akismet/
JetPack is a collection of tools that are maintained by Automattic who are the same people that develop WordPress.com. I don’t recommend automatically installing this plugin, but if there are features in JetPack that you need/want, it’s a good alternative to third-party plugins because you know they’re built on quality code and will be maintained and supported for time to come. Check it out at https://jetpack.me/ and http://wordpress.org/plugins/jetpack/. If you use JetPack, don’t go nutty and enable every feature; just enable what you need to minimize the overhead.
An Analytics Package
Use something to track your visitors and engagement. JetPack has a good stats tracking package built in, but you can also use something like Google Analytics or another third-party tool. I’m not going to recommend anything specific, so just pick one that you know how to use and interpret and go with it.