dd-wrt is a third-party (not the router's manufacturer) firmware, which was originally develop based on linksys WRT54G firmware. maybe thats why there's the 'wrt'. however i dunno what the 'dd' stands for. here's for further info.

well, i'm not a dd-wrt sifu. but to me, dd-wrt is essentially a firmware that will transform ur cheap, ordinary router to a stable, feature rich one. for instance, these r the things that i like in my dd-wrt (there r many other, however not within my interest), which can be hardly found on most home-class routers.
- nocatsplash : force everybody using internet from ur router to view a certain page 1st.
- support hotspot system : chillispot, wifidog, sputnik.
- static dhcp - pre-set the dynamic ip for ur users based on their mac. by using this, it is easier to control access, usage etc for different kind of users (based on ip range).
- mac filter - up to 128 users (most stock firmware supports <30)
- ur router can operate in multiple modes: ap, wds, client bridge, repeater bridge. these modes r useful if u want to expand ur wireless network
- universal repeater mode - can repeat any wifi signal, without any configuration (or knowledge) on the original signal source
- tweak ur router - increase transmit power (to extend range), overclock the cpu, select which antenna to use etc
- port forwarding
- QoS
- ddns
- url filtering

i bought a dir-300 yesterday and have successfully flash it to dd-wrt v24 rc5. its kinda difficult if u r a noob. check the guide here.

however for routers like linksys wrt54gs/gl etc, u can flash to dd-wrt using the same way as upgrading ur firmware.