The size of a fat quarter will depend on were you buy it. An English fat quarter is a quarter of a meter, where as an American fat quarter is a quarter of a yard. Always check the packaging for the size, especially if you have a particular project in mind.
I mainly use fat quarters for Patchwork, which originally developed as a thrifty craft, using fabrics recycled from old clothing and other items. If the finished design is going to be washed a lot , or example a baby quilt, then the same type of material needs to be used due to the different washing temperatures i.e. all cotton, however any type of fabric can be used for decorative projects like wall hangings.
Fabric can be bought for a project, will come in various sizes and designs, usually colour coordinated. Fat quarters are precut amounts, like jelly rolls.
The best way to choose which type of fabric to use is to decide what the finished project will be. babies cot quilt need to be soft and durable fabrics that are washed frequently. If it is a wall hanging then the fabrics could be more specialized. I choose my fabric by starting with a patten I love, then I match off that to create the project I am making.
All fabric should be washed before use in case they shrink and some of the colours can bleed together.