Q:

What are the different grain directions, and how do they affect joint strength? 

What are the differences between edge grain, face grain, and long grain (and why not throw end grain in there while we're at it)? 
And which of these grain directions results in the strongest glue joints? 

A:

Edge and face gain are both longitudinal grain, where grain runs along the surface.  Long grain = longitudinal grain, just less of a mouthful, much easier to type quickly too! 
The structure of wood, its grain, can be visualised as being like a bundle of straws tightly packed and bonded together.  In longitudinal grain we're seeing the side of the straws.  
End grain is anywhere where the cut exposes the ends of the straws.  So the two ends of nearly every board has exposed end grain.  Many diagonal cuts in wood also expose end grain, but the shallower the angle of the cut the more it involves longitudinal grain (this is why very shallow angles are chosen for scarf joints, to maximise glue surface area as well as increase the innate strength of the glue joint). 
There are in essence three glueing situations in woodworking:

long grain bonded to long grain
end grain bonded to end grain (as in a classic mitre joint)
long grain bonded to end grain

Long grain | long grain
All long-grain to long-grain joints (regardless if face to face or edge to edge, or a combination of the two as in some 90° joints) should be the strongest possible glue joint, if done well exceeding the strength of the wood.  So a fairly well-worn adage of woodworking "creates joints stronger than the wood itself" can literally be true[1].  
The glue industry's own testing, independent lab tests and any well-run test done in the workshop will invariable show this, where if a piece is broken apart you can expect that splits will largely happen in the wood and will essentially ignore the joint lines[2].  
End grain | end grain
End-grain to end-grain joints are the weakest glue joint.  They can be made much more secure by very good surface preparation and by sizing the end grain ahead of final assembly, but in general for any joint requiring good strength simply glueing together end grain won't be strong enough[3]. 
Visualising the grain in wood as being like straws and end grain as being like the ends of these straws helps explain why it bonds poorly normally, because those open ends of the straws are very absorbent and capillary action transports glue away from the surface, leaving too little behind to form a good bond. 
What sizing does is seal or partially seal the straws, preventing over-absorption of the glue. 
Long grain | end grain
Long-grain to end-grain joints are naturally somewhere in between the above two.  They were/are considered fairly weak, and woodworking guides will nearly often say that they need to be reinforced in some way — by nails, screws or dowels.  
Without reinforcement they can be too weak to rely on in isolation, but, if done properly (which would involve sizing the end grain prior to final glue-up) they can actually be strong enough in some situations without any mechanical reinforcement, e.g. because other joints in the piece will help hold things in place and everything mutually supports itself.  
[1] This is sometimes erroneously expressed as "glue is stronger than the wood itself" which is not strictly accurate.  In general it is only the glued joint that is stronger, NOT the glue, and it's important to bear this in mind so that you don't make the mistake of expecting glue to support gaps "because it's strong" (usually it's not).  Epoxy if one of the few glues commonly used by leisure woodworkers where you can rely on it to fill gaps strongly. 
[2] Because this is so universally shown to be true it can be used as an acid test of glueing procedures by any woodworker.  If in their testing the glue joints fails they can be confident that either they've done something wrong, or the glue is at fault.  Possible causes of a weaker long-grain to long-grain joint include:

The wood not being smooth enough — rougher surfaces don't normally bond well with most glues used in woodworking, a notable exception being epoxy.  As a rule you want very smooth surfaces on your wood for the best glue joint, smooth but not burnished. 
The joint faces are not flat enough, e.g. the angle is off so the edges don't make good contact along their entire length. 
Insufficient clamp pressure.  With most wood glues you want to clamp hard enough that all excess glue is squeezed from the joint, which in practice means you want to clamp very firmly. 
Glue is too thin. 
Glue is too thick (too dry). 
Glue has has 'gone off' in some way, by being frozen or because it's aged. 
Not enough glue was applied (an excess of glue should always be applied for maximum bond strength).  
Another cause is just a failing of procedure, so that even with perfectly jointed wood and new glue you don't get a good joint.  The classic reason for this is where glue was applied but then the joint wasn't brought together in time, so the glue dried out too much to form an adequate bond (either to the other piece of wood or to the glue already applied to the other piece of wood). 

[3] This is not to say that end-grain to end-grain joints are never strong enough.  The classic small box has mitre joints on all four corners and because of the small size and the lack of stress on the structure of the piece these joints can be perfectly acceptable and are in fact widely used.  But once a box needs to hold any major weight or needs to be robust enough to withstand lots of movement or rough handling then either the mitres need to be reinforced (e.g. with a spline) other joints should be used instead (e.g. finger joints/comb joints, box joints or dovetails).