Can you give more context to your "active hydrogen"? What were you reading about when you found it?
I don't know of any specific definition of active hydrogen, but I have heard it used loosely in at least two different contexts. One is the number of acidic hydrogens in a compound, typically in small peptides. -OH hydrogens, -NH hydrogens, -COOH hydrogens, -SO3H hydrogens, any hydrogens that can be pulled by a reasonable base in water (for example NaOH). Compounds were describes as having "molecular weight X with Y active hydrogens".
There is also a use I've seen in polymer chemistry, where prepolymers are prepared with molecular weights in the 100-1000 range, then reacted with other prepolymers or initiators to form the full length polymer. Most recently, a polyurethane preparation starting with isocyanates. In that case, "active hydrogens" on the prepolymer meant sites that could react with the isocyanate. This gives you a measure of how much branching the polymer will have, and how solid a three dimensional matrix will form. If there are only two active hydrogens, the chains can only be linked end to end, and any branching would have to come from the isocyanate piece. If there are more active hydrogens, each will reacti with isocyanate and form a more heavily branched and therefore more rigid final product.
Then there are a number of pseudoscience applications which have already been alluded to, where "active hydrogen" means as little as "phlogiston".