There is a minimum weight for the cars (including driver). Ballasts can be added to reach the minimum weight, which must be maintained at all times throughout the duration of the grand prix weekend. There's some controversy about the minimum weight limits being too low and favouring the lighter drivers. Hulkenburg being one of the heavier drivers reportedly missed out on a drive at Mclaren for reasons of weight and the difference it makes. 1 kilo of weight equates to around 0.035 seconds per lap on average for most circuits. A comparison between Alonso and Hulkenberg for example, shows (for their respective weights) 68 v 74 kg, this difference equates to around 0.2 secs/lap if the cars weigh the same. It's easy to see how this a significant time difference over race distance - 50 laps being in the region of 10 seconds.
Edit: It's important to remember developing the car throughout the season invariably means adding weight (new parts), which means there is increased pressure on the drivers to lose weight further to compensate (Most teams operate as close to the weight limit as possible for the obvious reason to gain faster lap times).
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