I do not understand the

acid + base ->conjugate base + conjugate acid

Please explain it to me with an example. I really do not understand it and I truly need your help.

I do not understand how he hydrogens transfer and stuff like that.

Your help is very much appreciated!

Thanks

The one with more H atoms in the acid; the one with fewer H atoms is the conjugate base.

H2AsO4^- + H2O ==> H3O^+ + HAsO4^2-
H2AsO4^- has more H atoms than the HAsO4^2-; therefore, the first is the acid and the second one the conjugate base.
For the H2O/H3O^+ pair,
H2O is the base and H3O^+ the conjugate acid. H2O has the fewer H, H3O^+ more. It's really that simple.