Drum hardware is a broad term, and could mean anything from...

Bass Pedals

The bass pedal is what you use to hit the bass drum. It is operated using one foot. Bass pedals come in two varieties - single and double. Some drummers use two bass pedals (on either one or two bass drums) for increased speed. Pedals come with drum kits, but on budget kits the hardware is not going to be sturdy.

Cymbal stands

These come in many varieties. You can get boom cymbal stands and straight stands, the boom having more angle possibilities. These hold the cymbals usually (not directly) over the drummer. The Cymbal is held in place with usually screw-on plastic ends and felt pads. To prevent the cymbal from brushing against the rod which goes up through the centre hole of the cymbal and causing 'key-holing' cymbal sleeves are put in place. Key-holing wears away at the centre of the cymbal through metal-on-metal contact between the stand and the cymbal, slowly making the centre hole bigger. This is NOT good at all and you may end up having to pay up for new cymbals (if you want to play with cymbals, that is).

Cymbal sleeves

The least known but one of the most important things on a drum kit. Cymbal sleeves are there to prevent the cymbal from brushing against the cymbal stand's rod which goes up through the centre hole of the cymbal and causing 'key-holing'.
Key-holing wears away at the centre of the cymbal through metal-on-metal contact between the stand and the cymbal, slowly making the centre hole bigger. This is NOT good at all and you may end up having to pay up for new cymbals (if you want to play with cymbals, that is).

Hi-hat stands

Now these are what hold your hi-hats up and hold a pedal which you press down on to open and close the hi-hats.

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