Here's a list of online tools that I use:
Open Site Explorer - Great for getting rich and valuable information about a domain. They provide information about where the links are coming from, their anchor text, how diverse they are and a bunch of other cool stuff.
SEOGadget - A great way of quickly looking at backlinks pointing to a website and providing you with a quick overview of which links should be removed. It's great to get a good understanding of backlinks and what to look out for.
Serpfox - SEOMoz provides a great rank tracker, but I find serpfox to be so much easier to track rankings and they update daily so you get a more up to date feedback. They also check rankings of up to 500 places. Your site is useless down there but it helps to gain a greater understand of where you rank and if you actually rank. Easy to quickly look at how well a website is doing as well.
Ahrefs - Another backlink checking tool. However, this one claims to update their index every 30 minutes. They constantly crawl the internet and that explains why their premium accounts are pretty expensive but the free accounts are useful enough. They show when backlinks have dropped, when new ones have been added, where they're from etc etc. The interface isn't as great as OSE but it's a useful tool to have a quick insight into what's happening.
Linkstant- This a new addition to the tools and something I've just set up on my own blog. Much like Ahrefs in terms of speed of link discovery, this one actually sends you an email when new links are found pointing to your website. Completely free, easy t set up. It took me two minutes to get things running. You can actually have an SMS sent as soon as a new link is found pointing to your website. It might not be as deep and thorough as Ahrefs but it sure as hell keeps you motivated to get those links! Also, unlike Ahrefs, you don't run out of credits.
Link Detective - A great tool for anyone that is serious about beating competitors. Basically, download a CSV from Open Site Explorer and throw it into the system. It will then tell you the websites makeup of links; directory, sitewide, blog comments, forums and so on. It also tells you the anchor text percentage as well. So if your website has been hit by a penalty on Google, run it through this. Any anchor text, that isn't you brand or naked URL, that is over 20% needs fixing. Diversify your links! It also tells you which links are dead, too.
These two tools require you sign up to SEOMoz. You can take a free 30 day trial which can be cancelled at any time, so give them a try!
SEOMoz Keyword Difficulty - If you're looking into a keyword and need to decide how difficult it is, then this is a great tool. They provide a percentage of difficulty for the keyword along with visitor numbers for this keyword. This information is found within Adwords keyword tool, but it provides a much easier way of determining just how difficult a keyword is. It's also an excellent way of comparing keyword difficulty and it's easy to then delve even deeper and find out which websites rank for a particular keyword.
SEOMoz On-page SEO - SEOmoz provides a great tool here. Enter your keyword and web page you're looking to rank and it will give you a run down and a grade of how well the website is optimised. They tell you if you need to add more keywords into the document, whether the title tags need tweaking and so on.
Edited by Jason Dexter, 28 November 2012 - 11:23.
Added Linkstant, Link Detective - 28/11/2012