While there are literally dozens of choices out there for compilers and IDEs for ARM, we're partial to GCC, which tends to have particularly good support for ARM chips in general.
While most seasoned veterans probably prefer a good old makefile, if you're new to ARM you'll appreciate the hand-holding an IDE can provide. We'll be using NXP's free LPCXpresso IDE in this tutorial, which is based on Eclipse and GCC, and works with all of their recent chips.
If you don't wish to use LPCXpresso, most of these steps also apply to a vanilla Eclipse + GCC installation, but you will have to do a bit more work yourself since LPCXpresso allows you to cut a lot of corners, generating the makefile and linker script on the go, etc. If there's enough interest, we'll also put together a tutorial on creating a makefile and building from the command line for this chip using only vanilla GCC.
Downloading the LPCXpresso IDE
In order to download NXP's free LPCXpresso IDE, you do need to create an account on Code Red's website (who are now owned by NXP), at https://www.lpcware.com/lpcxpresso/download
While registering sucks, LPCXpresso is still the easiest free tool using an open source toolchain, and you can easily break out of it once you're a bit more comfortable with ARM and GCC, creating your own makefile and linker script, and LPCXpresso is just a bit of icing on top of the Eclipse + GCC cake.
An advantage of LPCXpresso is that it includes a relatively easy to use installer for Linux, Windows and Mac OS X, so you're free to use the toolchain and IDE on any major platform.
After registering, you should see a download list similar to the following:
These are the latest downloads for Windows, but LPCXpresso is also available for LInux and MAC OS X.
The installation process is relatively straight forward, and it should be fairly foolproof. Starting with the welcome screen, just run through the installer, and LPCXpresso will automatically setup all the tools, including the cross-compiling toolchain, Eclipse, etc.
At the end of the installation process you can optionally look at some documentation on how to use LPCXpresso by selecting or deselecting the checkboxes, but at this point you've installed everything you need to start writing programs for the LPC810 (or any other LPC chip from NXP)!
Next you can start the LPCXpresso IDE (you may need to activate it if this is the first time using it), and you'll get a screen similar to the following: