TITLE: Coset and ideal
QUESTION [5 upvotes]: Up to this moment I still have fuzzy ideas about some definitions in Abstract Algebra, especially about the difference between coset and ideal. Perhaps this question is a dumb one for you but I know precise definition plays a very crucial role in study of this math. Any plain English explanation would be much appreciated, especially any links to intuitive visualization of the ideas.
Thank you very much for your time and help.

REPLY [4 votes]: There is not a direct analogy between a coset and an ideal - there's nothing "similar" to them. 
There is a much closer analogy between an ideal in a ring and a normal subgroup in a group. A normal subgroup is a subgroup that has additional closure properties; not every subgroup is normal. An ideal in a ring is a subring that has extra closure properties; not every subring is an ideal. 
These objects are important for quotient groups and quotient rings. Given a group $G$ and a subgroup $H$, you can form the quotient group $G/H$ if and only if $H$ is normal. Given a ring $R$ and a subring $S$, you can form the quotient ring $R/S$ if and only if $S$ is an ideal.