So, an acid is defined by how the molecule creates hydrogen ions, which are individual molecules in a fluid. A strong acid like HCl (hydrocloric) "dissociates" and creates a lot of hydrogen ions H^+ and chloride ions. Cl^- A weak acid, like vinegar creates fewer.
Now the acid breaks things apart by reacting with them. So if you have something like a glass test tube, the bonds between the silicone in the glass is way stronger than the hydrogen bonds, so it doesn't interact.
Now take something like a biological creature. The hydrogen ions get in and mess with the cell membranes (phospholipid) they mess with the way proteins fold, causing denaturation (unfolding or changed shape). The chemical bonds holding your skin together are fairly strong, but not strong enough to resist hydrogen ions.