I think that (one of) my favorite things is Sylow theory/ nilpotent groups. More specifically showing that nilpotent groups are the largest set of groups which behave as abelian groups do with respect to Sylow theory.
I think the fact that I'm finishing my undergraduate studies and remember in my first years battling away with small abelian groups and the like trying to find out if they are normal or if they have elements of certain order having these results so that these questions become easy is an example of where I can do things that I did not used to be able to do (easily).
Then they also have applications in classifying groups and Galois theory (which I am learning about just now), so it seems these guys are pretty useful!
I'm not sure I'm allowed to pitch in twice (especially when I'm just making suggestions of things I like as opposed to my "favorite things" but oh well...)
I have just been doing a little bit of reading for my final year dissertation which very roughly is on foundations of mathematics and was looking at some independence results and got slightly distracted on a forum and came across a pretty incredible independence result, the following statement is independent in ZFC:
If $X$ and $Y$ are sets and $X$ has cardinality greater than $Y$ then the number of subsets of $X$ is greater than $Y$ or If $\kappa,\lambda$ are cardinals and $\kappa>\lambda$ then $2^\kappa>\2^\lambda$
That's just crazy for me! I think my favourite part of mathematics is that I just don't understand it and I can just come across results all the time that blow me away!