We bought a Fuji (Fujipix) compact digital camera for my wife last year and she's very pleased with it. It does everthing my wife wants for photos taken on holiday, at the zoo, in museums and indoors at home. It cost £120 and has a x5 zoom lens.
There are so many compact digital cameras around at a wide range of prices. I would recommend that you pay a medium price. Don't buy a real cheapy for £60 or so as you get what you pay for. You can pay £400+ but I can't see much difference over a £120 compact.
One word of caution Yelenots! Don't buy a compact camera that runs off 2 AA batteries. They don't last long and you will be forever changing batteries. When you settle on a camera, make sure it has lithium-ion batteries as one will usually last all day. My wife's cost £5 each on-line and we have four.