You have found a better role, been through the interviews, accepted an offer and resigned. Then your employer responds with more money, a new title or promises about the future. It is flattering, and it deserves proper thought rather than a quick answer either way.
Work out what has changed
The counter-offer arrived because you resigned. The useful question is which of your reasons for leaving it fixes. If you were leaving purely over pay, a strong counter-offer may solve the problem. If you were leaving over progression, project quality, workload or how the commercial team is treated, a pay rise changes none of those things.
Judge the offer on specifics
A good employer countering to keep someone they value is acting reasonably, and staying can be the right decision, particularly if the offer includes specific, dated commitments. A named role in September at a stated salary is a real offer. A general promise that things will improve is not.
As with any contract, the commitments that matter are the ones in writing.
Think about the relationship afterwards
Some employers accept a resignation and retraction without holding it against you. In others, you may quietly be treated as a flight risk when promotions or sensitive projects are handed out. Judge your employer by how they have treated other people who almost left, rather than by how the retention conversation felt at the time.
Remember the role you would be giving up
In a stressful resignation week, the new role can fade into the background, even though it was appealing enough to interview for and accept. Bring it back into the comparison, and look at where each option would put you in three years rather than just at the two salaries this month.
Where we stand
As recruiters we clearly have an interest in this decision, so we will be plain about our approach. If staying is the right decision for you, we would rather you stayed. When a counter-offer arrives we will talk it through with you on its merits. What we discourage is deciding by default, because the reason you looked elsewhere was a future your current role was not offering.