By fritter, do you mean the kind that is deep fried - I assume so, if you are going to use beer batter. If so, you can make a batter using the following ingredients:
1 3/4 cups self-raising flour
1 good pinch of salt
a generous amount of ground black pepper
1/2 tsp paprika (optional)
1 cup beer
100 ml water (approx)
Mix all the dry ingredients in a bowl. Then make a well in the middle and add the beer and water. Mix well then let stand for an hour until it froths up. All you have to do then is dip sliced potatoes into the batter and deep fry. Keep the slices in a bowl of water until you are ready to dip, so they don't go brown. I've used cauliflower, sweet potato and mushrooms, too. I've also done curried ones (a bit like pakora) by chopping the veg a bit smaller, adding some cumin and putting some curry powder in the batter.
For potato cakes you basically mash potatoes then add stuff like leeks, spice, onion etc, a bit of flour and some beaten egg - then shallow fry.
Alternatively, go for a rosti-style thing of grated potato shaped into patties and shallow fry.