Liabilities are basically things that you owe to other people. Usually that’s some kind of debt, but it could be taxes or even equity that’s owned by someone else (minority interest).
(Incidentally, a car is an asset. The loan balance on the car, or, more accurately, the present value of the payments you will make to pay off the loan, is a liability.)
For banks, the main liabilities are deposits (money owed to depositors), regular bond-like debt, deferred taxes, and perhaps legal settlements. For things like pension funds or endowments, liabilities are expected payouts to plan participants or to the entity that the endowment supports.
Sometimes it’s easier to think of liabilities as if you are short a bond. If you owe a payment in X years, it’s like being short a zero coupon bond that matures in X years.
So what happens in banks is that they generally try to make money by lending over the long term at a high interest rate, and pay their depositors at a lower short-term rate. They can also try to do something similar with the bond liabilities where they argue that they lend out at high rates to risky projects, but bank bondholders get a more diversified portfolio that is safer and therefore get paid less. Banks make money on the spread, which is what the equity holders in banks get excited about.
Here’s the issue. If liabilities have a short duration (because deposits are more short term) and the assets tend to be long-duration, because they are being lent out long term at a spread. Then the assets can have a higher interest rate sensitivity than the liabilities. If interest rates rise, then both assets and liabilities are worth less than before, but the assets (i.e. loans to people who borrowed from the bank) take a bigger hit than the liabilities do.
If a bank hasn’t managed their assets and liabilities right, this means that a bank that used to be solvent, suddenly isn’t anymore, because assets that used to be worth more than the liabilities are suddenly worth less than the liabilities.
The other places where liability management is extra important are in pension funds, and corporate treasury. Usually, what one does is start by trying to set up fixed income portfolios that match the cash flows and timing (or at least the duration) of the liability streams, and then depart from that mix based on how much risk one is willing to take of not being able to meet liabilities (or how much of a cushion one has before not being able to meet them).