Traditional finger foods that most people think of include soft fruits like banana slices [cut in half to avoid a choking hazard], mango, soft peaches, pears, etc., cheese, soft-cooked vegetables and crackers. These make good dips.
But you can also be adventurous with your children! Try dipping a sandwich – and make it fancy by cutting it up with a fun-shaped cookie cutter. Or try dipping different shapes of pasta into tomato sauce.