They re quite simple, really, though like anything they can be a bit opaque to a newbie.
First, the "LOC" part. Line Of Credit. The lender agrees to lend to the borrower up to the limit of the line, any time during the period of the line. There is no repayment requirement until the end, when any outstanding balance must be paid or refinanced. (Edit: Typically the borrower must pay interest monthly but this can be done by borrowing against the line.) I had a line that had seen no activity, zero balance, for years. I viewed it as a standby resource. Some will report that LOCs have been canceled or limited by lenders from time to time but I have no personal knowledge of that.
Second, the "HE."Home Equity is the most common collateral for a consumer loan. For businesses the collateral is typically all of the business's assets. The collateral for a consumer doesn't have to be the HE if the borrower has other collateral that the lender will accept. I have seen adverts from Schwab saying that they will loan against assets in taxable accounts. Terms, rates, etc. I have no idea.
Since an LOC may be open for years it is common for the interest rate to float on an index. "One over prime." "Two over LIBOR." Both Prime and LIBOR have become somewhat discredited in recent years, so our most recent HELOC renewal floats on some other index but I don't remember details. Interest rate risk is always there for a long loan and someone will always bear it. A HELOC with a floating rate basically means that the borrower is bearing the rate risk.
The only other thing to understand is that for consumer loans, lenders like standardization in all aspects. Documentation, collateral, etc., so you are unlikely to get a loan collateralized by your pristine 1948 Willys Whippet or based on your daddy's promise to include you in his will.