As said above, American DVDs are Region 1, while in Europe and the UK we are Region 2!
(The region coding was invented by the US Movie studios so they could control when films were released to DVD in different areas of the world, and so maximising earnings for the films in the cinemas).
If you have a 'multiregion' DVD player then you should be able to play region 2 DVDs. Although some DVD have added protection called 'RCE' or Region Code Enhancement. This can cause problems on some multiregion players.
You also need to check that your TV and DVD can handle the American TV format - NTSC (we use PAL in the UK). Most modern DVDs and TV can play either format.
Having said all that, there isn't that much to be saved money wise by buying in the US. You get better deals buying on-line from sites like PLAY or CD-WOW.
On of the advantages of buying DVDs in the US is that some titles are released in the US earlier than in the UK. Also, some of the extras may differ on the US version to the UK version.
If you read some of the DVD review sites, some people think the sound quality can be better on US DVDs due to the different scan rates (60Hz instead of 50Hz), not sure on the technicalities of this. I can never tell the difference
US CD's should have no problems playing in thep UK, although some albums may have a slightly different track list on the US version than on the UK version.