HC stands for "high credit" on a credit report and refers to the spending limit you have on each of your credit accounts.
Your high credit limit is used to determine your debt-to-limit ratio, which is how much debt you are carrying in relation to how much debt you are permitted to carry.
Your debt-to-limit ratio has a substantial impact on your credit score. The FICO credit scoring formula factors in your debt-to-limit ratio when determining your overall credit score. If your debt is too high in relation to your limit, it will hurt your credit score. That in turn will hurt your ability to get credit on favorable terms.
You may call your creditors at any time and request a higher credit limit. If you are approved, the "high credit" notation on your credit report will update to reflect the new limit.
Going over your credit limit will leave you with a high debt-to-limit ratio and damage your credit score. It is also likely to result in fees from your creditor.