To Construct a dispenser you need 7 cobblestone, 1 bow, and 1 redstone.
Redstone can be found very deep underground and looks like a stone block with red dots in it. Cobblestone is very common - just mine the gray blocks you see. Finally, the bow has to be crafted from three sticks and three pieces of string. String can be obtained by killing spiders.
Once you have all of the items, open the "Weapons" tab of the inventory and craft your bow, then tab across to the "Redstone" tab in your inventory and create the dispenser.