The typical acid, for instance a hydrogen halide, has a polar covalent bond. That is a bond (i.e. #H-X#), where the electron density is strongly polarized to the more electronegative atom, the halogen. A #H-X# bond, while strongly polarized, retains some covalent character, and indeed #H-Cl#, and #H-Br#, and #H-I#, are gases at room temperature, with discrete #H-X# molecules. When you open a bottle of conc. hydrochloric acid sometimes you can inadvertently get a snort of #HCl#, (open it and use it in the fumehood, Bozo!).
Of course, in water, the #H-X# bond can become so polarized that we get discrete #H^+# and #X^-# species. These are aquated species, i.e #H_9O_4^+#, a cluster of water molecules with an extra #H^+#; this is why we often represent the acidium ion in water as #H_3O^+#.
On the other hand, aqueous bases, #KOH(aq)#, or #NH_3(aq)#, contain atoms with 1 or more lone electron pairs, that can bind #H^+#. If there is a query, or something you would like to ask, you are welcome to ask for clarification.
