What are sibilants and velars, and why are they important?

I've noticed these linguistic terms relating to consonants come up regularly in explanations of the Russian spelling rules, so I'll try and explain very roughly what they are.

Sibilants are sounds that you make with air from your vocal chords hitting your teeth. In Russian, these are:

ж, ч, ш, щ

Velars are sounds you make at the back of your mouth with your tongue against the roof of your mouth. In Russian, these are:

г, к, х

Wikipedia has two much more detailed articles about sibilants and velars, which may be slightly baffling to anyone who hasn't studied linguistics (that includes me!).

Why are they important?

These are the letters that break the normal spelling conventions. See this lesson on Russian spelling rules.