Ionizable hydrogen atoms for acids
Not all hydrogen atoms in a compound are equally acidic. The acidity of a hydrogen at a certain position in a molecule depends on its chemical environment, i.e. the solvent (if the compound is dissolved), which atoms and functional groups it is attached to, and how well the negative charge of the corresponding conjugated base is stabilized. A quantitative measure for the acidity of a proton is its $pK_a$ value which is defined as follows (reference):

$$pK_a=-\mathrm{log}_{10}K_a$$

with $K_a$ being the equilibrium constant of the dissociation reaction in water $\ce{HA + H2O \rightleftharpoons H3O+ + A-}$, in which the hydrogen is abstracted from the compound by heterolytic bond cleavage. $[\ce{H2O}]$ can be assumed to be constant (approx. 55 mol/l) and is therefore included in $K_a$:

$$K_{a}=\frac{[A^{-}][H^{+}]}{[HA]}$$

Let's take formic acid ($\ce{HCOOH}$) as an example. As the name of the compound suggests, it has at least one acidic proton, and if all hydrogens would be equally acidic and dissociate from the molecule, the reaction would look like this:

$$\ce{HCOOH + 2H2O \rightleftharpoons 2H3O+ + COO^{2-}}$$

However, this does not happen in aqueous solution, only one of the protons is acidic:

$$\ce{HCOOH + H2O \rightleftharpoons H3O+ + HCOO-}$$

The reason for this is that both hydrogens are situated at different positions in the structure:



With a $pK_a$ of 3.77, the hydrogen of the carboxyl group is significantly more acidic than the hydrogen bonded to the carbon atom. The typical $pK_a$ of an aldehyde hydrogen (H bonded to the carbon of a carbonyl group) is in the range of 17 (source), so such a position cannot be deprotonated in water.