"Burning" in the common vernacular, is simply reaction with oxygen.
Hydrogen reacts with oxygen to form water. It has a reasonably high ignition temperature that typically requires at least a spark to ignite, but it gives off a lot of energy when it burns.
H2 + O2 ---> H2O
The "exhaust" is the gas that comes off from the burning process. Assuming that the hydrogen was being burned in a car engine, that exhaust would contain water from the reaction, plus all of the excess atmospheric gases that went into the engine, plus any atmospheric gases that were formed due to the high temperature of the engine (typically reactions of nitrogen with oxygen).
Normally our car engines are fueled with gasoline, which is a mixture of hydrocarbons (hydrocarbons are compounds which contain only hydrogen and carbon atoms). These hydrocarbons also react with oxygen - the carbon reacts with oxygen to form carbon dioxide (CO2) and the water reacts with oxygen to from water.
CxHy ---> CO2 + H2O
The exhaust contains carbon dioxide and water, plus all of the excess atmospheric gases that went into the engine, plus any atmospheric gases that were formed due to the high temperature of the engine (again, typically reactions of nitrogen with oxygen). However, if there isn't enough oxygen to completely convert all of the carbon to carbon dioxide, you will also get some carbon monoxide formed. Also, since the gasoline is coming from crude oil and is not a pure gas the way hydrogen is, there is frequently also at least a small amount of sulfur present which will burn to form sulfur dioxide.
We actually do know what the exhaust in both cases is, in great detail under a wide variety of possible conditions.