You can use use any of the Shimano UN series bottom brackets.
Such as UN26, 53, 54, 72, 73, etc.
Generally the higher the number then the higher the quality.
You have the main dimension (110) and, as Chainbrain says, you need to know the bottom bracket width (68 or 73). (There is a third size - 70mm - but this is the Italian type and you are very unlikely to have this)
Two ways of determining the size =
1) There is usually a label on the cartridge that states 110 x 68 (maybe 68x110).
2) Measure across the BB shell in millimetres.
To change the BB is quite straight forward (as long as no one has used threadloc) but you do need the correct tools.
"Crank Puller" and "Cartridge tool".
See:
http://bicycletutor.com/cartridge-bottom-bracket/
But remember the drive side is "left hand threaded"