What is a inner hull on a submarine?

There are 2 hulls on every modern submarine; the outer hull, known as the superstructure, and the inner hull, known as the pressure hull. The best analogy is that it's like a thermos bottle, the only difference being that instead of air between the inner and outer portions, there's water.

The superstructure is essentially for streamlined submerged travel while underwater, and to house masts and other equipment (line lockers, cleats, MBT valves, topside under-ice sonar fairings, sonar arrays, etc.). The pressure hull on most boats isn't really designed for underwater travel so much as it is to house equipment and crew, and the design itself is spherical/cigar shaped to give it the best pressure resistance.

The inner pressure hull is where the main systems and crew reside. The hull for most boats (save for Russia and China, who use titanium and GRP) are made of durable, flexible steel that compresses and expands with changes in sea pressure as a boat moves through different depths. Hull penetrations from the inner to outer hull (valves, masts, etc.) are carefully inspected and tested before each major operation, and at intervals during the year. One method of sealing everything (or finding problems) is taking a boat to test depth, which is above the crush depth of a submarine. The sea pressure forces tight any seals that may be a bit loose.

Of course if there's something wrong, a catastrophic failure is also possible.