The fundamental relationship to keep in mind is that bond strength is inversely related to bond length. A stronger bond is shorter. The problem arises in how bond strength is determined. When we say a bond is strong, we refer to the bond having a high bond dissociation enthalpy (BDE), that is, the energy input required to break it. A strong bond is harder to break, and requires more input energy, thus has a higher BDE.
For an acidic hydrogen, we observe that the hydrogen is given up easily, implying that the bond to it's conjugate base anion is weak. Consider acetic acid:
We know the acid hydrogen is more acidic than the alpha hydrogens, because the conjugate base of acetic acid is acetate, which has lost the proton from the breakage of the O-H bond. This would imply that the O-H bond is weaker than the alpha C-H bonds. However, when BDE is considered, we see that the O-H bond should be stronger than the C-H bonds. Why is a "stronger" bond broken more easily?
The explanation is that acidity is a function of what's called heterolytic bond cleavage. Acidic H's leave as protons, right? They take no electrons with them; those remain with the conjugate base. In contrast, BDE values are derived from homolytic bond cleavage reactions, where the electrons in the bond are split evenly.
In general, more polar bonds are stronger, because the bonding electrons are held more tightly at one end of the bond. This means to redistribute those electrons evenly throughout the bond so it can then be broken homolytically would require more energy than to just break a bond which already has evenly distributed electrons (i.e., a nonpolar bond). The O-H bond in acetic acid is more polar than the alpha C-H bonds, meaning it's harder to break homolytically, and thus has a higher BDE. However, it's much easier to break the O-H bond heterolytically than it is to do the same to the C-H bonds. See the deal?
So if you're referring to bond strength in the context of acidity, you have to be careful how you specify the type of bond cleavage and what measure of "strength" you're using.
With regard to hydrogen bonding, involvement in a hydrogen bond draws the partially positive hydrogen further away from the F/O/N that it's bonded to, and thus lengthens and weakens that bond.
*Note that for carboxylic acids, acidic loss of the acid hydrogen is more thermodynamically favorable than acidic loss of the alpha hydrogens due to resonance stabilization of the oxygen anion, so it's not simply bond polarity contributing to the site of deprotonation. Still, acetic acid provides a simple example for this explanation. One could also just look at ethanol, though we don't consider that as readily acidic as a carboxylic acid.
