Fill Down and Fill Right are essentially shortcut methods for copying contiguous cells. Cells may contain numbers, dates, or formulas.
Filling is simpler than copying and pasting because it requires only one selection and one operation (Fill) rather than a selection, a Copy operation, another selection, and a Paste.
Because it uses only one selection, however, Fill can only be used when the paste-to area lies adjacent to (either to the right or below) the copy-from area.
With Fill Right, the cells of the single leftmost column that you've selected are copied into all the selected cells directly to their right.
With Fill Down, the cells of the single topmost row that you've selected are copied into all the selected cells contiguously below them.
The following animation demonstrates how Fill can be used to copy dates and formulas.
Notice how Excel recognizes month names and "does the right thing," or increments the month name for each column (Jan, Feb, Mar,...).
Fill is useful because often you'll lay out a two-dimensional block of contiguous cells containing formulas that are nearly the same, differing only by cell references within them. Because relative cell references in formulas are adjusted when replicated (with either Fill or Copy/Paste), you need only figure out the details of a formula in one cell, then replicate it throughout the block of cells that are to behave similarly.
When you are copying formulas you should understand the important related concept: