What sort of blood pressure problem requires the use of an ACE inhibitor?

Any, if tolerated. Ace inhibitors are a cornerstone of almost any hypertensive patient, especially if a patient has diabetes or heart failure. Monitoring of kidney tests and potassium and watching for cough or rarely rashes or swelling of the lips and eyelids are adverse effects that need medical attention.
High blood pressure. This class of medications is used for hypertension as well as for heart failure.

Related Questions

Have any undesirable effects been seen in long term use of the blood pressure drugs of the ACE inhibitor class?

Same as short-term. Analyses have not shown any clear risks, including with regards to risk of cancer. Risks over long-term appear to mimic that seen in short-term use (e.g. Risk of hi potassium, kidney function abnormalities); sometimes this can show up after years on the medicine. There may even be benefits to chronic use (apart from intended benefits), such as lower risk of diabetes or atrial fibrillation.
Early side effects. The potential side effects, if they're going to occur, such as cough, hyperkalemia, and renal impairment - all occur early on. If you've tolerated the medicine for a few weeks, you're going to be fine over the long term.