While it doesn't specifically relate to all cases of in-game slow down, the term LAAAG!! is screamed by all those affected. How badly your client will lag behind events in the game world depends on several factors.
When you initially log into the game, the shards are listed according to their ping. This is a basic measurement of how long it takes for your client to send information to that server, and then for that server to send a reply back to you (very similar to the "ping" used in sonar). New players should typically select the shard with the lowest ping to ensure the best game play. Usually these servers will be closest location-wise to the player in the real world. The closer your house is to your ISP, and the closer your ISP is to the game server, the lower your ping will be - Some players enjoy pings as low as 20ms or less.
Using your internet connection for other tasks (eg. web surfing) while playing limits the amount of information your game client can send to/receive from the game server at once, effectively increasing ping time.
You'll also find that what you encounter in the game world will affect performance. An area filled with many players, monsters, and items requires that a lot more information be passed through your connection - effectively increasing the amount of lag you'll see. Large custom houses in particular can slow the game quite noticeably when going close to them for the first time, more so if you actually enter (this causes the client to load all the items stored within).
Slower computers sometimes have trouble loading new sound effects etc as you play. Turning off unnecessary cosmetics can sometimes speed things up in these cases. The Kingdom Reborn client has a zoom feature that can be activated using the mouse wheel, pulling the view inwards decreases game lag while reducing visibility.
- Ultima Online - Latency - Official Site's UO Trace tool.