Since one of my previous job responsibilities was figuring out how to protect aircraft from the effects of lightning, I have a pretty good idea of what happens.
Lightning is best described as "pulsed DC". The leader that initially establishes the discharge path is, in simple terms, a static charge which has reached sufficient potential to ionize a path through the air. This ionized path is what eventually carries the actual lighting discharge. There is very little current associated with the leader, and as far as how lightning affects people, buildings, etc., the leader can be ignored.
Once an ionized path is established, the part of the discharge that we see (and that creates thunder) occurs. This is an initial high current pulse, called the first return stroke, which may reach 100,000 amperes in a severe strike. This current decays rapidly (on the order of milliseconds)to a much lower level. Most of the actual charge transfer occurs during this longer lower level discharge, which is limited by how fast the charge can move to replace what was originally discharged. If 100,000 amperes isn't current, I don't know what is!!!
Sometimes the initial discharge is followed by repeated discharges when enough charge flows into the vicinity of the ionized path before the path dissipates, and creates a second (or third, or fourth, etc.) flash. This phenomenon is why lightning sometimes appears to flicker - it really is a series of discharges.
In a single strike, current only flows in one direction. Cloud-to-ground strikes generally originate in a pocket of negative charge in the cloud, i.e., the cloud is negative with respect to the earth. Cloud-to-ground strikes can also originate in a positively charged area of a cloud, in which case the earth is negative with respect to the cloud. A positive charge is simply a lack of electrons, a negative charge is a surplus of electrons. In a cloud-to-cloud strike, the leader can initiate in either a positive or negative pocket of charge.
I've attached a link to a NASA website with a good discussion of lightning.
To answer the original question, yes, lightning is in fact a very large pulse of electrical current. Any individual strike is DC, as charge only moves in one direction. A single lightning strike may contain multiple discharges, but they will all be of the same polarity. Other strikes may be the opposite polarity, but the polarity does not change during an individual discharge.
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