What are the differences between rings, groups, and fields? Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used?
They should feel similar! In fact, every ring is a group, and every field is a ring. A ring is a group with an additional operation, where the second operation is associative and the distributive properties make the two operations "compatible". 

A field is a ring such that the second operation also satisfies all the group properties (after throwing out the additive identity); i.e. it has multiplicative inverses, multiplicative identity, and is commutative.
Every field is a ring, and every ring is a group. A group has one operation which satisfies closure, associative property, commutive property, identity, and inverse property. A ring satisfies all properties of a group; it also has a second operation which has closure, associative, and distributive property between these two operations. A field has two operations and both satisfy all group properties.