You can use arrow keys or the mouse.

Left-Click on pixels to draw (or press SPACE).
Right-Click to select the color under the cursor (or press ENTER).

Left click on a color in the palette at the bottom to select it. (ALT + arrow keys will pick palette colors too!)

See the "Undo" bar along the right? Click there to revert to an earlier state, or press CTRL+Z to go back one step, so to speak.

On the left is a panel of tool buttons. Each of them can also be triggered with a keypress.

D Draw tool. Use this for single
  pixels or free-hand lines.

B Box tool. Use this to draw
  filled rectangles.

L Line tool. Use this to draw
  straight lines.

F Fill tool. Use this to flood-fill
  paint an area. Like the paint
  bucket in Photoshop.

R Replace tool. Use this to replace
  one colour with another.

O Circle tool. Use this to draw non-
  filled circles. The first click
  defines the center of the circle.

A Airbrush tool. Use this to scatter
  pixels around. While the airbrush
  is active, you can press + and -
  to change the scatter area, and
  CTRL+ and CTRL- to change the
  "mist", which is the number of
  pixels scattered with each click.

S Scroll tool. While this is the
  selected tool, you can click and
  drag on the tile, or press arrow
  keys, to move it around. Pixels
  will wrap over the edges.

M Mark tool. Marks a rectangular
  area of the tile to use with the
  clone tool.

C Clone tool. Draw on the tile using
  a portion of a tile that you have
  previously chosen with the mark
  tool. Pixels drawn in color 0
  (normally black) will be
  transparent. While you are using
  the clone tool, you can hold the
  ENTER key or the right mouse button
  and move the center of the cloned
  area

Backspace flips the tile horizontally.

CTRL+Backspace flips the tile
  vertically.

[ Rotates the tile 90 degrees
  counterclockwise.

] Rotates the tile 90 degrees
  clockwise.

CTRL+S saves the tile immediately without exiting, which is useful if you are live-previewing the game.