It could be several different errors but this one is the most likely.
In your counter-strike or cstrike directory you should have a liblist.gam, have a look with notepad to check if the line:
gamedll "addons\metamod\dlls\metamod.dll" is there.
If it is then change it to:
gamedll "dlls\mp.dll"
save the liblist.gam and start a new cs1.6 game. Once in the game choose team and stuff and once you are able to run around shut down cs1.6 and change the liblist.gam back to:
gamedll "addons\metamod\dlls\metamod.dll"
This has to be done every? time steam is updated.
If this doesn't help we'll try some other correction
(It would be nice if you told us how you installed it, what dirs and what files you downloaded and so on....)