What characteristic of a base enables it to ''accept'' Hydrogen Ions?

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LogicalAcid
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According to the Brønsted-Lowry definition, a base is a compound that can -keyword- accept H+, and an acid anything that can give H+.
 
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Someone please clarify this for me.

And also, here is something

HCl → H+ + Cl− ( I am guessing the arrow is referring to before and after being put in water)


The addition of H+ to the hydroxide ion (OH−), a base, produces water (H2O), its conjugate acid:

H+ + OH− → H2O ( and I am guessing this is after it ionizes and reacts with water?)
 
Be careful how you label your species here. OH- can be seen as a base (and a nucleophile) but H2O is not the conjugate acid in the sense that you mean, H3O+ would be the 'acid'. This is because (using your HCl equation) HCl does not simply become two separate ions in solution. The acid will protonate H2O to become H3O+.

But remember, all these reactions are in equilibrium and the pKa determines to what extent the different species exist.

H20 <---> H+ + OH-

HCl <---> H+ + Cl-

H2O + H+ <---> H3O+