thenApply and thenCompose are methods of CompletableFuture. Use them when you intend to do something to CompletableFuture's result with a Function.
thenApply and thenCompose both return a CompletableFuture as their own result. You can chain multiple thenApply or thenCompose together. Supply a Function to each call, whose result will be the input to the next Function.
The Function you supplied sometimes needs to do something synchronously. The return type of your Function should be a non-Future type. In this case you should use thenApply.
CompletableFuture.completedFuture(1)
.thenApply((x)->x+1) // adding one to the result synchronously, returns int
.thenApply((y)->System.println(y)); // value of y is 1 + 1 = 2
Other times you may want to do asynchronous processing in this Function. In that case you should use thenCompose. The return type of your Function should be a CompletionStage. The next Function in the chain will get the result of that CompletionStage as input, thus unwrapping the CompletionStage.
// addOneAsync may be implemented by using another thread, or calling a remote method
abstract CompletableFuture<Integer> addOneAsync(int input);
CompletableFuture.completedFuture(1)
.thenCompose((x)->addOneAsync(x)) // doing something asynchronous, returns CompletableFuture<Integer>
.thenApply((y)->System.println(y)); // y is an Integer, the result of CompletableFuture<Integer> above
This is a similar idea to Javascript's Promise. Promise.then can accept a function that either returns a value or a Promise of a value. The reason why these two methods have different names in Java is due to generic erasure. Function<? super T,? extends U> fn and Function<? super T,? extends CompletionStage<U>> fn are considered the same Runtime type - Function. Thus thenApply and thenCompose have to be distinctly named, or Java compiler would complain about identical method signatures. The end result being, Javascript's Promise.then is implemented in two parts - thenApply and thenCompose - in Java.
You can read my other answer if you are also confused about a related function thenApplyAsync.