When a company wants to lay off an employee, how does it make this difficult situation more acceptable to the person being laid off?
It gives them money. This money is called 'severance pay'. As a part of the 'severance pay', a company legally has pay the employee; a payment for outstanding days of holiday and bonuses that are owed them, and a payment based on the number of years they have worked in the company (the more years, the higher the payment). In addition to this, many company will include in the 'severance pay' to the employee an extra 'thank you' payment.
Although this extra payment would seem like a nice gesture by the company, it is actually used to encourage the person to leave their job quietly or to force them to resign.
Depending on both the company and country (or state), the 'severance pay' may also include certain conditions that the employee has to be abide by if they accept the money (e.g. not to work for a competitor, take legal action against the company, share information about the company etc...).
Not everybody is entitled to severance pay when they leave a company. People who are fired/sacked for personal misconduct, have worked at the company for less than a year or resign by their choice don't normally receive it. These people will be paid for outstanding days of holiday and bonuses that are owed them, but nothing else.
Often when people are laid off/made redundant they don't only just receive money, but other things as well (e.g. private pension). This combination of 'severance pay' and other things is called a 'severance package'.
To learn more vocabulary connected to this topic, you can do a free online exercise on job related vocabulary.