One of the power of OpenStreetMap is that every one can do whatever they like with the data. There are actually lots of services provided by websites other than the main official one. Some consider this also a weakness because at first sight it seems OSM provides much less than other map services like Google, even if there are already some many functionalities available. Here is my list of the services I like most. Maybe it can point to some services that you didn't know :-)
http://www.openstreetmap.org : the official website, that everyone knows. The one feature I miss is a "Where Am I?" button to show my approximate position (using the browser known location). Many websites have it, but not the main OSM website! Confusingly, there is a "where am I?" link, but it doesn't show you where you are, it tells you the name of the current place on the map :-/
http://project-osrm.org/ : Open Source Routing Machine, a really fast routing website with a nice clean interface. It can find the fastest route from any where in the world in less than a second! I just hope they will eventually add support for bicycle routing, and it will be perfect :-)
http://maps.skobbler.co.uk/ : Skobbler, a very nice interface for simple user of the map. I especially like the way to search and see POIs.
http://simon04.dev.openstreetmap.org/whodidit/ : Who Did it? Shows the last changes on an area. Really useful to monitor the last changes over an area. It basically overcomes all the shortcomings of the history functionality on the official website.
http://keepright.ipax.at/report_map.php : KeepRight, a service that detects some potential errors in the data. Works world-wide and is updated about weekly. IMHO, the most useful warnings are "missing tags" and "almost-junctions". Helped me to fix so many errors already.
http://tools.geofabrik.de/osmi/ : OSM inspector, a similar service with different type of warnings detected.
http://osmbugs.org/ : OSM bugs, an interface where it's very easy to flag map errors. It's good to check it every so often around your area of interest to see if you can help fix some errors.
http://maposmatic.org/ : MapOSMatic, create your own paper map of any city you like.
http://polar.openstreetmap.de/antarctica/ : To see what you've never seen. The typical world projection only goes down to 85° latitude, and the areas around the poles are already very distorted. This shows Antarctica in an meaningful way.
http://hdyc.neis-one.org/ and http://yosmhm.neis-one.org/ : How did you contribute and Your OSM Heatmap: Show in a simple and good looking way the contribution of a user. In theory, it's good to know at a glance the level/contributions of a user, but let's admit it, the main usage is to boost your ego ;-)
That's the main websites I'm using. If you think there is some other useful ones that are not there, let me know!