Alfresco has grown into a large set of projects. What was once described as a document repository now includes web content management, Microsoft SharePoint and Office integration, a web-based collaboration environment (Share), CMIS capability, and a lot of frameworks you can use to extend the capabilities of the components.
It can be time-consuming and difficult to install and configure all of the components - even though it is well worth it. If you have a good understanding of what components you're interested in, it will be a lot easier. Take your time during the install, and take notes along the way about decisions you make. (Where you put things, what usernames and passwords are, and what versions the various components are).
You must have a JDK (Java Development Kit) available. It can be Java 5 or 6.
You must have a database. You can evaluate Alfresco with its built-in Derby database, but if you really want to use this, you must have something else. It's most likely better to start out with something like MySQL than to move to it later.
Some sort of Java application server. This can be Tomcat (recommended if you don't have a preference) or JBoss or many others. If you don't have an application server already, downloading the tomcat distribution instead of the war-only distribution is a lot easier as it puts things where they need to be.
What installation situation am I?
There are lots of things people might want to do with Alfresco. In all cases, you will be downloading the main labs release, but which scenario you want will determine what else you have to do.
You want only the document repository to store and manage documents. You want the web client to interact with it. In this case, you'll be working with a single war file (alfresco.war) and ignoring other components for now. (You may want to install modules later on)
Repository and Share
You want the repository on the back end, but you really want to try out Share - the new web-based collaboration environment that sits on top of the repository. You will care about the alfresco war and also the share war. You also should install the swftools (to get nice document previews) and also open office (so you get nice document thumbnails in share, so that the document preview works for office documents, and for document transformations in the repository). Installing open office takes a little time (it's easier in the all-in-one installers) and installing swftools can be difficult as it has its own dependencies and on Linux, you have to build it yourself. (Don't get hung up on this - you can add swftools later - which is probably a good idea unless you're on Windows and don't have to build it)
Repository, Share, and Microsoft Office/SharePoint integration
You will want to install the repository war (alfresco.war) the share war (share.war), open office, swftools, and the supporting SharePoint components (vti module and root.war).
You are an early-adopter, you want everything, and want to play with it
In this case, you need all of the above, you also need to grab the latest source out of subversion, you need a decent IDE to manage a really huge codebase (something like Eclipse or Netbeans) so you can make a little change here and there and test it out yourself. It's probably going to be too much to run it from an IDE, so install it normally, and have your subversion directories elsewhere. Make changes, build individual archives and swap them into your install if you need to.