What might be an even better question is How can I remember all of the different common algebraic structures? How do I keep track of all of the "different" definitions?
I use quotes because most of the definitions are very similar.
It is important that we understand homomorphisms : whenever someone says homomorphism they mean it preserves all of the structure that it can! (unit, product, sum, everything in sight)
Firstly two rules that you always have whenever you could have them (for the most part) :
1) associative law : whenever there is an operation it is associative, this is a rule that only involves one operation. Again, almost every algebraic structure has every binary operation being associative... except for non-associative algebras.
2) distributive law : whenever there are two operations there is the distributive law. I am pretty sure this one is always there.
Most basic :
Monoids (some people call them different things, and there are more basic notions, but i dont think they are actively studied in algebra. In fact, I dont think monoids are actively studied, this definition is handy though for later purposes) : You have one operation and a unit identity element for that operation.
Commutative monoid : like above but the operation is commutative.
Groups : a monoid with inverses FOR EVERYTHING!
Abelian Group : a commutative monoid with inverses FOR EVERYTHING!
From this point on everything is an abelian group plus some extra stuff.
Two operations :
these come in one of two flavors: another operation, or the action of something external on the guy you are looking at.
another operation (usually we think of one of the opeerations as additive and the other as multiplicative, this is common and don't let it scare or worry you) :
Rings with unit : these are monoid objects in the category of abelian groups... whatever that means. The right way to think about that last sentence is that there is some structure sitting on top of the abelian group structure, and they play nicely together (the distributive law! multiplication by a fixed element is a homomorphism of abelian groups). (after we understand the rules for rings with units we can think about rings without units... if we even want to worry about them now. Some people take ring to mean ring with unit, it really depends on the book)
Commutative Ring with unit : like a commutative monoid in the category of abelian groups right? yep... whatever that means. Mostly just look at the above. Looking a tthe commutative versions of things is usually an easy transition to make.
Division algebra : This is like a group object in the category of abelian groups except there can't be an inverse for zero, or rather the additive identity. So $R$ is a division algebra if $x \in R\setminus 0$ has an inverse for all $a \in R$. Note here $R$ must have a multiplicative identity. Essentially $R$ is a ring and $R\setminus 0$ is a group.
Field: Like a division algebra except commutative, so $R$ is a ring and $R\setminus 0$ is an abelian group.
External operation :
All of these are Modules in one form or another. Always.
Module: an abelian group with an action of a ring on it so that the ring action satisfies some distributivity condition.
Vector space: A module over a field.
Ideal: A submodule of $R$. Note that if $R$ is a ring than we can think of it as acting on itself via multiplication, this is the sense in which an ideal is a submodule.
Later I may come back and add examples if you like.
Let me know if I left something out that you want me to add. I strongly think the above intuitive definition is far better than memorizing the equations one needs. If you learn something about commutative diagrams then you oly need the diagrams, and there are only really 2 or 3 of those. Then you will see it really is always the same concept.