I suddenly felt dizzy when smoking a cigar the other day. What happened there?

That was a nicotine flash. It can be compared to drinking too much coffee in too short time. Both bodily reactions are basically identical with each other.
It's not dangerous, but absolutely unpleasant.

It's truly scary when you experience it for the first time. One feels cold and starts to sweat at the same time. It usually happens when the stomach is empty, depends on the personal day's form. And there of course individuals who are more sensitive than others. Experience also helps. The body gets used the nicotine and you will notice when a flash is coming sooner. So you can stop or reduce the puffing frequency.

It will go by after a couple of minutes. The best medicine is a standard syrup to cure stomach complaints.
The easiest solution is to eat some sugar and something that builds a solid food pad in your stomach. A good bread with some cheese on it and a cup of cola are perfect.