Acids are substances that release hydrogen ions easily. Hydrogen ions can react with many substances, including skin and flesh. When your skin comes in contact of an acid, the acid reacts with the skin and may destroy flesh, or cause heat burns, since the reaction releases heat.
Some acids have other additional unique properties that make them very dangerous. Nitric acid not only produces hydrogen ions, but also nitrate ions: this is a very powerful oxidising agent and can react in other ways, which causes very severe burns if the acid is concentrated. It can also react very violently with many organic substances, sometimes explosively (Pure nitric acid is used as the oxidiser in some rocket fuels!).
Sulphuric acid when dilute is not very dangerous. It can cause irritation/burns. But when concentrated, suphuric acid is a very powerful dehydrating agent. It is so good at absorbing water, that it can remove hydrogen and oxygen from other molecules to form water! Many organic substances consist of carbon-oxygen and hydrogen, when they are exposed to concentrated sulphuric acid, the hydrogen and oxygen are removed to form water, leaving carbon. This reaction is very exothermic, so in addition to reacting chemically and dehydrating your flesh/skin, it can cause thermal burns from the heat released.