Firstly and most importantly do not provide links or addresses to sites that provide roms. It breaks the forums AUP. Please delete the reference from the above post.
Secondly dependent on what games you are trying to play you may need additional files.
Some games have several 'sets' these are different regional versions of the game. Only one of these sets contains all the game files, this is called the 'parent' rom with the other sets just containing the files that are different for the different versions.
You can find out which are parents by looking at the mame32 game list. There is an entry on the list that lists whether the game is a clone, (clone of) there will be one entry for each game that has no clone info, this means it is the parent rom.
Some other games need a bios to function (like neogeo games). You need to download the bios (like neogeo.zip) and put it into the mame32 rom folder the same as your other game roms. You only need to do this once for each game system that needs one, so for example once you have downloaded the neogeo bios and put it in the rom folder, all neogeo games will work with it.
There are also a number of games that require the game rom and the game CHD (compressed hunks of data) file, this is a large file some of which are over 1Gb that contain large amounts of information stored in the original arcade machine either on a hard disc, cd-rom or other storage medium.
Lastly your games must be compatible with the mame version you are running. Most sites will state which version of mame the roms are compatible with. In the case of the unmentioned site you posted they usually keep their roms fairly up to date.
Hope that helps.
If you are having issues with a specific game then post the game name and the romname and we'll take a look.