Increasing amounts of research are revealing that crows (and other corvids) are incredibly intelligent.
However, how exactly does one define "intelligence"? Well, it's difficult, but one way that it is measured, especially across different species, is flexibility or adaptability. This can include tool use, because technically speaking, tools themselves are an adaptation.
Much of the intelligence that crows exhibit seems to have to be related to their capacity for memory. For example, they are able to not only remember specific humans, but they can also hold grudges, as evidenced by a study where people wearing masks captured and tagged some crows. The crows, even years later, still react negatively towards the masks.
But they can also communicate knowledge to each other somehow, because all those years later, there were crows who did not have direct experience with the masked tagger/capturers, yet still attacked those who wore the masks. So, this is an obvious bonus as far as intelligence goes–knowledge gained by one bird can be passed to another, and so on.
There have been other cases where crows appear to be putting their skills of memory and communication to work. For example, they will change their migration patterns if crows have perished in certain places before.
They can also use tools, which is a classic indicator of “sophisticated levels of cognition”. There have been many studies involving crows using tools, and in fact, crows are capable of not only utilizing tools, but also of MAKING them–specifically hooks, a behavior which has been observed in both laboratories and the wild. In the wild, the crows have fashioned hooks out of sticks and leaves and then proceeded to fish for insects. A pretty similar occurrence happens in laboratories; generally, the experiments involve food that is only obtainable through the utilization of tools, and the crows do not disappoint.
Another ability born of crow’s memory is the ability to plan. They can not only remember information but process the information and adapt their behavior accordingly. For example, in a recent experiment, it was discerned that ravens (of the same family as crows, called "corvids") can plan for the future as well as human children can.
One final fun fact about crows is that they are able to understand the concept of death, at least on an elementary level. Perhaps their common association with death is a fitting one.
Historically (i.e., in myth in fable), people seem to have been on to something when it comes to crows (and everything else I guess, I mean the reason they hang around is that they continue to have relevant meaning). One of Aesop's fables details a thirsty crow who drops pebbles into a jug of water so that the water levels will rise enough for it to be able to drink it. The fable is purportedly emphasizing the "virtue of ingenuity", which sounds a lot like flexibility or adaptability, aka those criteria of intelligence.
Additionally, given the ability of corvids to remember, Odin (hailing from Norse mythology) was perhaps on to something when he named his messenger ravens "Huginn" and "Muninn", which respectively translate apparently into "thought" and (evidently the second is harder to directly translate) something like "memory", "desire", and/or "emotion". At first glance, memory seems unrelated to desire and emotion but I suppose one would need a certain amount of memory to be able to develop something like emotion or desire. Or perhaps I am just trying to back up my own opinion. In any case, crows, ravens, corvids–they're pretty darn smart.