This list is from the back page of the official rules (with a few tweaks reflecting our implementation). It pretty much summarizes everything you need to know about how an order will be adjudicated.
More detailed discussions can be found in the "Common Questions" subforum.
1. All units have the same strength.
2. There can only be one unit in a province at a time.
3. Units of equal strength trying to move to the same province cause all those units to remain in their original provinces. (This is called a standoff.)
4. A standoff doesn’t dislodge a unit already in the province where the standoff took place.
5. One unit not moving can stop a series of other units from moving.
6. Units can’t trade places without the use of a convoy. (Units of equal strength trying to trade places without a convoy causes those units to remain in their original provinces.)
7. Three or more units can rotate provinces during a turn provided none directly trade places.
8. A unit not ordered to move can be supported to hold in its province. (The support adds to its strength.)
9. A unit ordered to move can only be supported by a support order that matches the move the unit is trying to make. (Again, the support adds to its strength.)
10. A dislodged unit can still cause a standoff in a province different from the one that dislodged it.
11. A dislodged unit, even with support, has no effect on the province that dislodged it. (It cannot cause a standoff there.)
12. A country can’t dislodge or support the dislodgment of one of its own units, even if that dislodgment is unexpected. (You can attack or support an attack on your own unit, but if that attack or support would normally cause the target to be dislodged, it won't be.)
13. Support is cut if the unit giving support is attacked from any province except the one where support is being given.
14. Support is cut if the supporting unit is dislodged.
15. A unit being dislodged by one province can still cut support in another.
16. An attack by a country on one of its own units doesn’t cut support.
17. A dislodgment of a Fleet necessary to a convoy causes that convoy to fail.
18. A convoy that causes the convoyed Army to standoff at its destination results in that Army remaining in its original province.
19. Two units can exchange places if either or both are convoyed. (This is the exception to Rule 6.)
20. An Army convoyed using alternate convoy orders reaches its destination as long as at least one convoy route remains open.
21. A convoyed Army doesn’t cut the support of a unit supporting an attack against one of the Fleets necessary for the Army to convoy. (This supersedes Rule 13.)
22. An Army with at least one successful convoy route will cut the support given by a unit in the destination province that is supporting an attack on a Fleet in an alternate route in that convoy. (This supersedes Rule 21.)