What do each memory_order mean?

Question

The GCC Wiki gives a very thorough and easy to understand explanation with code examples.

(excerpt edited, and emphasis added)

IMPORTANT:

Upon re-reading the below quote copied from the GCC Wiki in the process of adding my own wording to the answer, I noticed that the quote is actually wrong. They got acquire and consume exactly the wrong way around. A release-consume operation only provides an ordering guarantee on dependent data whereas a release-acquire operation provides that guarantee regardless of data being dependent on the atomic value or not.

The first model is "sequentially consistent". This is the default mode used when none is specified, and it is the most restrictive. It can also be explicitly specified via memory_order_seq_cst. It provides the same restrictions and limitation to moving loads around that sequential programmers are inherently familiar with, except it applies across threads.
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From a practical point of view, this amounts to all atomic operations acting as optimization barriers. It's OK to re-order things between atomic operations, but not across the operation. Thread local stuff is also unaffected since there is no visibility to other threads. [...] This mode also provides consistency across all threads.

The opposite approach is memory_order_relaxed. This model allows for much less synchronization by removing the happens-before restrictions. These types of atomic operations can also have various optimizations performed on them, such as dead store removal and commoning. [...] Without any happens-before edges, no thread can count on a specific ordering from another thread.
The relaxed mode is most commonly used when the programmer simply wants a variable to be atomic in nature rather than using it to synchronize threads for other shared memory data.

The third mode (memory_order_acquire / memory_order_release) is a hybrid between the other two. The acquire/release mode is similar to the sequentially consistent mode, except it only applies a happens-before relationship to dependent variables. This allows for a relaxing of the synchronization required between independent reads of independent writes.

memory_order_consume is a further subtle refinement in the release/acquire memory model that relaxes the requirements slightly by removing the happens before ordering on non-dependent shared variables as well.
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The real difference boils down to how much state the hardware has to flush in order to synchronize. Since a consume operation may therefore execute faster, someone who knows what they are doing can use it for performance critical applications.

Here follows my own attempt at a more mundane explanation:

A different approach to look at it is to look at the problem from the point of view of reordering reads and writes, both atomic and ordinary:

All atomic operations are guaranteed to be atomic within themselves (the combination of two atomic operations is not atomic as a whole!) and to be visible in the total order in which they appear on the timeline of the execution stream. That means no atomic operation can, under any circumstances, be reordered, but other memory operations might very well be. Compilers (and CPUs) routinely do such reordering as an optimization.
It also means the compiler must use whatever instructions are necessary to guarantee that an atomic operation executing at any time will see the results of each and every other atomic operation, possibly on another processor core (but not necessarily other operations), that were executed before.

Now, a relaxed is just that, the bare minimum. It does nothing in addition and provides no other guarantees. It is the cheapest possible operation. For non-read-modify-write operations on strongly ordered processor architectures (e.g. x86/amd64) this boils down to a plain normal, ordinary move.

The sequentially consistent operation is the exact opposite, it enforces strict ordering not only for atomic operations, but also for other memory operations that happen before or after. Neither one can cross the barrier imposed by the atomic operation. Practically, this means lost optimization opportunities, and possibly fence instructions may have to be inserted. This is the most expensive model.

A release operation prevents ordinary loads and stores from being reordered after the atomic operation, whereas an acquire operation prevents ordinary loads and stores from being reordered before the atomic operation. Everything else can still be moved around.
The combination of preventing stores being moved after, and loads being moved before the respective atomic operation makes sure that whatever the acquiring thread gets to see is consistent, with only a small amount of optimization opportunity lost.
One may think of that as something like a non-existent lock that is being released (by the writer) and acquired (by the reader). Except... there is no lock.

In practice, release/acquire usually means the compiler needs not use any particularly expensive special instructions, but it cannot freely reorder loads and stores to its liking, which may miss out some (small) optimization opportuntities.

Finally, consume is the same operation as acquire, only with the exception that the ordering guarantees only apply to dependent data. Dependent data would e.g. be data that is pointed-to by an atomically modified pointer.
Arguably, that may provide for a couple of optimization opportunities that are not present with acquire operations (since fewer data is subject to restrictions), however this happens at the expense of more complex and more error-prone code, and the non-trivial task of getting dependency chains correct.

It is currently discouraged to use consume ordering while the specification is being revised.

This is a quite complex subject. Try to read http://en.cppreference.com/w/cpp/atomic/memory_order several times, try to read other resources, etc.

Here's a simplified description:

The compiler and CPU can reorder memory accesses. That is, they can happen in different order than what's specified in the code. That's fine most of the time, the problem arises when different thread try to communicate and may see such order of memory accesses that breaks the invariants of the code.

Usually you can use locks for synchronization. The problem is that they're slow. Atomic operations are much faster, because the synchronization happens at CPU level (i.e. CPU ensures that no other thread, even on another CPU, modifies some variable, etc.).

So, the one single problem we're facing is reordering of memory accesses. The memory_order enum specifies what types of reorderings compiler must forbid.

relaxed - no constraints.

consume - no loads that are dependent on the newly loaded value can be reordered wrt. the atomic load. I.e. if they are after the atomic load in the source code, they will happen after the atomic load too.

acquire - no loads can be reordered wrt. the atomic load. I.e. if they are after the atomic load in the source code, they will happen after the atomic load too.

release - no stores can be reordered wrt. the atomic store. I.e. if they are before the atomic store in the source code, they will happen before the atomic store too.

acq_rel - acquire and release combined.

seq_cst - it is more difficult to understand why this ordering is required. Basically, all other orderings only ensure that specific disallowed reorderings don't happen only for the threads that consume/release the same atomic variable. Memory accesses can still propagate to other threads in any order. This ordering ensures that this doesn't happen (thus sequential consistency). For a case where this is needed see the example at the end of the linked page.