Predicate logic is what you get when you add quantifiers to propositional logic. With propositional logic, it is part of "first order logic". You can think of predicate logic as just the various rules governing quantifiers, in which case predicate logic + propositional logic = first order logic, or you can think of predicate logic as being the quantifier rules added on to propositional logic, in which case predicate logic is more or less the same thing as first order logic.
Quantifiers are ∀ - the Universal quantifier (an upside-down "A", for "all"), and ∃ - the Existential quantifier (a backwards "E"), "Some". This allows you to make more general claims- with propositional logic, you could say If A then B. But with predicate logic, you can say things like "For all x, if A is x, then A is B".
It is interesting to note that the Universal quantifier does not suppose the existence of the quantified: It is correct to say "∀x" (For all X......)when the number of x is zero. However, "∃x" (For some x.......)is more accurately translated as "There is an x such that x........"
With the rules of inference, there are some interesting things that you can start doing with predicate logic, such as Robinson Arithmetic.