The difference between Func and Action is simply whether you want the delegate to return a value (use Func) or not (use Action).
Func is probably most commonly used in LINQ – for example in projections:
list.Select(x => x.SomeProperty)
or filtering:
list.Where(x => x.SomeValue == someOtherValue)
or key selection:
list.Join(otherList, x => x.FirstKey, y => y.SecondKey, ...)
Action is more commonly used for things like List<T>.ForEach: execute the given action for each item in the list. I use this less often than Func, although I do sometimes use the parameterless version for things like Control.BeginInvoke and Dispatcher.BeginInvoke.
Predicate is just a special cased Func<T, bool> really, introduced before all of the Func and most of the Action delegates came along. I suspect that if we’d already had Func and Action in their various guises, Predicate wouldn’t have been introduced… although it does impart a certain meaning to the use of the delegate, whereas Func and Action are used for widely disparate purposes.
Predicate is mostly used in List<T> for methods like FindAll and RemoveAll.