They are numbers for every stat that are determined when you first meet the Pokemon.
Each stat's IVs can go to a maximum of 31, with each of those 31 numbers being worth one point in the stat it is for. So a Pokemon could have 6 Attack IVs, 15 Defence IVs, 31 SAtk IVs, etc.
This isn't the only thing IVs do. They also determine the characteristic your Pokemon has depending on which stat has the highest IVs (the things like "Proud of it's power" and "Loves to eat"), but this does not mean that the stat that the characteristic matches has 31 IVs. It could have as low as two with the other stats having just one IV. They will also determine which type of Hidden Power the Pokemon has if it learnt the move and how much Base Power it'd have, but there's a load of math behind that.