Some of the big problems (single corps attacks, random DOWs) have been discussed elsewhere, and I'm sure will be hashed out in great detail in this thread. Here are some smaller but still significant problems:
1) Not thinking far enough ahead. For instance, AI Russia usually declares against Sweden in Jan, and sends its army into Finland. Once Finland falls, it sees that it needs to get into Sweden to prevent lapse of war, so it sends an army across the northern frontier. But then supply all the way to Stockholm is too expensive that way, so it just sits there until eventually it is allowed to starve, and the war lapses. Result: if I get control of Sweden, I can always keep it.
2) No backup plan. Example: When AI Turkey invades Egypt, it sends a force south into Sinai, and then tries to move around the Egyptian army to reach Cairo without risking a field combat. But if you place the Egyptian army in the area that blocks access from Sinai to Egypt, the Turks still go south, then sit there, waiting for you to clear a path. If you just sit and wait long enough, they eventually starve. Again, result: if I get control of Egypt, I can always keep it.
3) Too amenable to informal peace. For example: playing France or Prussia or some other far-away country, I may have manipulated alliances in the Caucasus. Russia declares war, giving me all those minors as free states. Russian army marches in, with no possible opposition anywhere in the region. Yet if I offer an informal peace, even though they have expended PPs to declare war, and will shortly be able to conquer the minors, they will often accept peace. Result: free minors for me, loss of PP for them. Sometimes you can quickly build your empire just by doing this repeatedly on the frontiers of an indecisive AI.
4) Unnecessary foraging. If the AI has to move its forces quickly--to defend the capital, for instance--it will often not bother creating a depot where it's going, even though it's within its own territory. In fact, it will create a depot the next month, having already incurred huge foraging losses.