The lowercase d command is like the lowercase c command in that it requires a motion.
The only difference really is that unlike the c command, the d command wont put you in INSERT mode.
Another difference is that the d command puts what you delete in a register. Sorta like your paste history.
The d command is use a lot with the v command and p command. With the v command puts you in VISUAL mode, and in this mode you can select text, hit the d key, and go to a different location in the text editor and hit the uppercase P to "put" or paste. This is basically Vim's cut and paste.
Note: If you have a paragraph that has no line breaks, and you hit the dd command it will delete the whole thing. In other words, it treats the whole paragraph as one line.
This will delete a character under cursor
This will delete the letter to left of the cursor
This will delete the previous word.
Note: I will only delete a whole word if you are at the beginning of a particular word. If you're in the middle of a word, it will delete only from the point of the cursor the the beginning of the word.
This will delete the word you are at, if the cursor is at the beginning of the word. If not, it will just delete to the end for the word.
This will delete a whole line
This will delete the line you're on plus one more up