Cardio is short for cardiovascular exercise which means really any exercise that benefits the heart. This used to be mainly aerobic training but now research suggests that weight lifting also benefits the heart - this is an example of a phrase that became popular before science could really validate it.

Aerobic exercise is any sustainable activity that utilizes oxygen, (and the burning of fat), for a primary source of fuel.

Now, this definition is muddied because you can have weight training, (i.e. longer sets that actually use oxygen and are therefore more aerobic, take circuit training, for example, when people aren't using a heavy load). You can also have what is traditionally thought of as cardio, (i.e. sprints that can be anaerobic depending on the rest. If the heart rate is allowed to lower between the intervals followed by maximum output then you are more in the anaerobic/glycolytic range). If you are only doing intervals but the heart does not drop substantially then you are going to be in more of a cardio range.

Basically, and this is a nutshell, there are primarily three systems that fuel energy. ALL energy systems are used to some extent, but various activity will use more of one or the other.

Anything lasting just a few seconds is going to be primarily anaerobic. This includes weight lifting, where the reps last a few seconds and are maximal bouts of output. Active rest occurs between the repetitions and then actual rest occurs between the sets. Shot-put, 50 yard-dash, weight lifting, olympic lifting, power-lifting are all examples of this.

Anything lasting a few minutes is primarily the glycolytic system where lactic acid is utilized to generate energy. This would be prolonged bouts such as sprints, longer weight lifting sets, etc. Basketball (run down court, then stay at the court = submaximal bout followed by active rest), and Soccer are examples of this.

Anything beyond that first few minutes must utilize oxygen turnover and therefore you are in the aerobic zone. This would be longer runs, rowing, hiking, etc.

 Keep in mind there is no black and white with this - you still cross over into multiple systems. The reason why HIIT is more cardio than, say, a 20 minute weight training session, is that in HIIT on the slower intervals you are still performing work. Your heart rate drops somewhat but doesn't drop a lot and therefore is still elevated in its output - this means that oxygen must fuel the process.

On the other hand, in typical weight training, you are resting between sets which allows adequate recovery for the heart rate to slow while the muscle energetics replenish themselves. A long weight training session with short rest and a lot of supersets is going to be aerobic at the same time as being anaerobic during the sets - you are anaerobic to explosively move the weight but because you are not allowing your heart rate to drop significantly between bouts of work the entire workout becomes more aerboic. Dave Draper was a huge advocate of this method of training and felt you could build muscle and stay lean or even burn fat simultaneously by supersetting a lot.