What is a strong acid called that has a lot of H+ ions?

Strong acid or base means it ionizes completely in water. It has nothing to do with concentration or corrosiveness. There are seven strong acids:

Sulfuric
Nitric
Hydrochloric
Chloric
Perchloric
Hydrobromic
Hydroiodic

It's easy to remember the list because there are three of the most common industrial acids, three "chloric" acids, three "hydro" acids, and they all make a clumsy "7" on the periodic table.

There are dozens of strong bases and hundreds of "almost strong" bases.

The concentration of H+ ions is called pH and you need to study for a while to understand that system. You have to take chemistry classes for a year and a half just to learn the words before they try to teach you these things.