Links to other information

Help Forum
Eve has the most newbie friendly community I've come across. You don't get bodyslammed for asking silly questions. Come hit the New Citizens forum and ask questions. Reading the golden rules thread is a good start.

Having difficulty knowing if you'd be better off with a PDU or a magstab? Wondering why forum posters are keen on getting IBTL or QFT? Not sure why canflagging in a nosdomi means wtfpwnbbq? Try the jargon guide.

Sites
The allabouteve site has a great list of all the other guides you can find. It's really worth looking at.

The official Knowledge Base is quite useful and answers many fiddly little questions.
The Eve Online forums search doesn't work properly. So, use Eve-Search instead. Seriously.

An NPC database (sort of like thottbot) is here. You can look up targets resistance, damage, bounty, etc.

Trading: this guide is excellent.

Mission agents: this search lets you find agents near your current station. It also works in-game!

Wiki: has excellent description of the racial ships, and more.
Grismar's Wiki is also comprehensive. There is no 'official' wiki, sadly.

This site has graphics and pictures of all the ships, I love it.

Electronic Warfare: wow, there is a lot to it, and this guide is superb! Learn how each race has different EW strengths, and how to counter them.

Bookmarks let you survive, hide and find things. Read more from Agony Unleashed's guide to bookmarks.

Scrapheap Challenge is a popular alternative to the official EVE forums, with a reputation for having good-quality information on PVP tactics and setups.

Screenshots of every ship
If you'd like to see all the ships you could possibly pilot, this brilliant site EveOnlineShips has screenshots of them all.

Applications
EveMon is an essential tool for all EVE players, and is used by the developers too. It has a lots of useful features, one key feature being the ability to pick a ship or a piece of equipment and having it automatically generate a list of the skills you need to train before you can use it (and how long it will take).
QuickFit is very useful for working out how you can fit out your ship before spending money on modules that you might end up unable to use. The 'capacitor simulator' feature and the tank analysis features are both very useful, too.
EVE Fitting Tool is an alternative to QuickFit - it's easier and more convenient to use, but doesn't have the powerful simulation and analysis screens. I think I'll probably use EFT to work out my initial fittings, and then fine tune them in QuickFit.
MLCalc is an essential tool for industrialists. Preloaded with every blueprint that can be built, it allows you to quickly find out how much it will cost you to build whatever you were thinking of building.