Mash, crumbles and pellets are the form the food comes in. Mash is very fine. Crumbles are small enough for chicks to eat, although adult chickens can eat it, too. Pellets are large and only the adults eat it. Chickens seem to waste less with pellets.
Starter, grower, start and grow and layer are different recipe formulations. Young chicks can eat either starter or start and grow. These foods have more protein in it, than food for an adult chicken, since fast growing chicks need more protein. Sometimes people switch chicks on starter to grower, part way through their growing period. Layer is for adult hens that are laying and has more calcium in it, to make egg shells.
Most often, you see chick starter or start and grow as crumbles. Layer commonly comes as either crumbles or pellets.
There are also whole grain or cracked grain feeds that are mixed up by feed mills or individuals that have the courser grain and a protein source, mixed with a powdery vitamin and mineral supplement.
Scratch is plain grains that are tossed out for chickens to scratch at and eat. It's not a complete diet and is lacking proteins, vitamins and minerals. It's usually cracked corn with various other grains or seeds mixed in. The combination used seems to vary by region.
