Not only in economics, but M1 and M2 are important to understand for the pure sake of consuming. M1 and M2, while both are representations of money, differ quite significantly.
I will start with M1. M1 is physical money supply. This can include any of the following: cash, coins, demand deposits, traveler's checks, and even checking accounts. M1's most important quality to understand, is the quickness and ease of transfer from accounts and assets to physical currency. Also known as liquidity, it is very easy for M1 to quickly become something you can hold in your hand. It is not "near money" as M2 might be.
With M2, not only does it include "near money," but it also includes cash and checking deposits. Near money can be looked at as anything from savings deposits, money mutual funds, and other time deposits that are less liquid and not easily transferable to physical money. M2 is never really used as a means of payment, that is all done in M1. However, M2 is important to understand because it can often serve as a legitimate way to invest, increase wealth, and even pay off loans, bills, and taxes.