Acids, bases, strong and weak acids

Acids are proton (H+) donors.
Acids can be cations (NH4+) or molecules (CH3CO2H, H2O) or anions (H2PO42–).
Bases are proton (H+) acceptors.
Bases can be either molecules (NH3, H2O) or anions (CH3CO2).

As indicated above, water can act either as an acid or as a base. H3O+ and OH are present in very low concentrations in pure water, due only to the reaction of water with itself.

Reaction 1:  H2O + H2 H3O+ + OH
 means that conversion of reactants to products occurs to an extent of less than 100%.

Species referred to as acids are stronger than water as their reaction with water to produce H3O+ occurs to a greater extent than Reaction 1. Thus aqueous solutions of acids have a higher H3O+ concentration than pure water.
Reaction 2:  

Species referred to as bases are stronger than water as their reaction with water to produce OH occurs to a greater extent than Reaction 1. Thus aqueous solutions of bases have a higher OH concentration than water.

Reaction 3:  Proton Transfer from water