Stream operations fall into two main categories, intermediate and terminal operations, and two sub-categories, stateless and stateful.

Intermediate Operations:

An intermediate operation is always lazy, such as a simple Stream.map. It is not invoked until the stream is actually consumed. This can be verified easily:

Arrays.asList(1, 2 ,3).stream().map(i -> {
    throw new RuntimeException("not gonna happen");
    return i;
});

Intermediate operations are the common building blocks of a stream, chained after the source and are usually followed by a terminal operation triggering the stream chain.

Terminal Operations

Terminal operations are what triggers the consumption of a stream. Some of the more common are Stream.forEach or Stream.collect. They are usually placed after a chain of intermediate operations and are almost always eager.

Stateless Operations

Statelessness means that each item is processed without the context of other items. Stateless operations allow for memory-efficient processing of streams. Operations like Stream.map and Stream.filter that do not require information on other items of the stream are considered to be stateless.

Stateful operations

Statefulness means the operation on each item depends on (some) other items of the stream. This requires a state to be preserved. Statefulness operations may break with long, or infinite, streams. Operations like Stream.sorted require the entirety of the stream to be processed before any item is emitted which will break in a long enough stream of items. This can be demonstrated by a long stream (run at your own risk):

// works - stateless stream
long BIG_ENOUGH_NUMBER = 999999999;
IntStream.iterate(0, i -> i + 1).limit(BIG_ENOUGH_NUMBER).forEach(System.out::println);

This will cause an out-of-memory due to statefulness of Stream.sorted:

// Out of memory - stateful stream
IntStream.iterate(0, i -> i + 1).limit(BIG_ENOUGH_NUMBER).sorted().forEach(System.out::println);