If you're asking how to emulate an art style, the answer can get a little technical. In simple terms, you need to examine how a particular artistic technique comes together and then figure out how to replicate that for yourself.
Would I be right in assuming you're talking about the in-game character sprites specifically? That uses a technique called pixel art, which essentially involves drawing at a low resolution with a sharp, non-antialiased brush. Tools like MS Paint and Paint.NET work great for this actually, others like GraphicsGale are made specifically for pixel art, though I personally developed my technique in Photoshop.
Pixel art can seem daunting due to its limitations, but i find it helps to sketch out what you're looking for using your usual techniques, and then reference that when creating pixel art. I usually take a large sketch, scale it down and trace the pose from that.
In fact, I just happen to have some backups of one particular piece of mine so you can see, vaguely, a character sprite in various stages. Don't let the gif fool you though, it's a process like any art is.
Oh, and
here's that sketch at full resolution so you can see what I started with. And here's the finished pixel sprite, alongside another I did with different limitations.
Other than that, the character sprites in SRB2 use a lot of the same artistic techniques as the character portraits, title screen and other artwork, and assuming that you drew your own profile pic, it seems to me like you already have a pretty good handle on SRB2's art style. The thick black outlines, the sharp lighting and shading, the bright colours. It's the same for the character sprites, just at a smaller scale.
