The size and type of weight to use is determined by quite a few factors;
The weight should anchor the tackle to the sea bed against the force of the current to stop it drifting off down the beach to the annoyance of others fishing nearby or dragging into snags. If there is little or no current such as in Poole Harbour or at Swanage then a small smooth weight will do the job. If the smooth lead is moved by the current then a watch lead (the circular lead with bumps on it ) will give a bit more holding power. In strong currents the grip lead is used, these have grip wires which are held in place by the springiness of the wire. A strong pull on the line will pull the grip wires out of the grooves that hold them and the weight can be reeled in.
The size of the weight will also be determined by the type of fishing you are doing and the safe casting weight of your fishing rod and of the line you are using. For fishing light tackle around piers and jetties you can use small smooth leads, for open beaches in rough weather you would have to use five or six ounce leads to get any hold in the sea bed. The flow of water and wind acts on the line and tries to move the weight, the finer the line the better in this respect and allows the use of lighter weights. Lighter weights are good for at least two reasons, you will be able to feel what the fish is doing as you reel in and a small lead is cheaper to loose than a heavy one.
Sometimes when trying to cast a big bait into the wind you will need to step up the weight to overcome the wind resistance. If there are big waves you may need a bigger weight as you can then tighten the line up more to stop the rod tip moving about as much, making it easier to spot a bite.On some sandy beaches you may get good results by using a ball weight and letting the current wash the lead around to attract fish such as flounder and plaice which are attracted by movement.
Lead is not the deadly poison that environmental fascists say it is; it has been used for millennia for fishing weights. To swallow lead is not a good idea, but then neither fish nor the angler is stupid enough to do that.... especially with the sizes of weights used in sea angling. Washing your hands before eating is a good idea, but as you have been handling smelly worms and fish you would do anyway.
I have fished a few times over rough snaggy ground with a character who, when waiting for a bite, looks around the beach for stones with a hole in them. He ties on a loop of line and uses them as weights, if he loses one, it's just going back to where it came from.
Weights are marked with a number of ounces or grammes, there are about twenty five grammes to an ounce for rough calculation, or about a hundred grammes to four ounces.
update Sept 2011
If you fancy making your own leads I have added a post here