Correct, liabilities are amounts owed by a person or business to another person or business.
Some examples of liabilities would be accounts payable (money owed to suppliers for good/inventories purchased on credit), mortgage loan (loan over property), personal loan, debentures, wages payable, unearned revenue and accrued expenses.
Classifying the liability into either current or non-current depends on it's lifespan i.e. the period over which the business or person intends to settle the amount owing. All amounts which will be settled within 12 months are classed as current liabilities. Any amounts that will be settled over a period longer than 12 months are classed as non-current liabilities.
As a rule of thumb, accounts payable, wages payable, unearned revenue, accrued expenses and any other expenses payable would generally be considered current liabilities. This is because they are mostly short term liabilities. Consider wages payable. staff members would generally not be too happy if their wages were only paid more than 12 months after they performed the work. Also, suppliers have set terms and conditions for payment of any goods bought on credit. This is usually around 3 months but depends on the contract.
This is not true for all cases and the underlying information and intention of all parties should be considered when determining the class. For example, suppose a company has recently been set up with a handful of family staff members. Their salaries are determined however they enter into an agreement with the company whereby their salaries are not to be paid out until the business is generating significant cash flow. They determine this to be more than a year after inception. In this case, wages payable would be considered a non-current liability.
On the other hand, items such a 20 year mortgage or a long term back loan would be classed as non-current liabilities.
To answer your second point on whether accounts can be classified as current and non-current the answer is yes. This is termed the "current portion of long term loan".
If you think about a 20 year mortgage, payments are made against this loan on a regular monthly basis. As such, the payments that will be made within the next 12 months are in fact current as they as due within that period. The remainder of the loan which will be paid between 12 months from now and the end of the loan is then the non-current portion. In this case, the outstanding amount is split and recognised partly as current and partly as non-current.