Let's look at the first two notes of the scale:

D is a whole step above C. You could also say that D is two half-steps above C. If you wanted to play in half-steps, you could play the C, then play the black note between C and D, then play D.

What's the black note between C and D called?

Now we get into sharps and flats. Please note that these don't necessary have to be black notes on the piano. Sharps and flats slightly raise and lower the pitch of a note.

C sharp (or C#) refers to the note that is one half-step higher than C. On the piano, this is the black note that is between C and D. On the guitar, this is the note that is one fret higher than whereever C is.

Where "sharp" refers to a half-step higher, "flat" refers to a half-step lower. You can see from the diagram below that "D flat" (or Db) and "C sharp" (or C#) both refer to the same note:

Well, that's confusing. Why have two names refer to the same note?

I agree that the whole flat/sharp thing can be pretty confusing in the beginning. Having these different notations come in super-handy when we're talking about different keys signatures, and the concepts of keys and key signatures come in VERY handy for songwriters. More on this later.

When writing music notation, the "flat" symbol looks like this:

The "sharp" symbol looks like this:

When indicating that a particular note is sharp or flat, put the sharp or flat in front of the note, like this:

You said that sharps and flats don't always refer to black notes on a piano. What gives?

Don't forget, a sharp raises a note a half-step and a flat lowers a note a half-step. So if look at an F on a piano:

And then lower it a half-step:

The more astute among you will undoubtedly notice than F flat (or Fb) refers to the exactly the same note as E.

By this point, I'm sure some of you are dying to practice writing out the notes you've learned as well as sharps and flats. You can buy manuscript paper in most music stores, or you can find staff paper online to print. Here is one good source of standard notation staff paper (from About.com).