What is a certificate in plain english? I've got this question from a course:

Show how to efficiently find a certificate for each of the following problems assuming that the decision problem is efficiently solvable.

And it seems like this has been poorly explained and I am getting conflicting answers on how to answer. For explanation purposes I will consider my problem to be f(x) where given a truthy value for x f(x) will output 1 and the opposite for falsey values.
In plain English, a certificate is a variable x such that the verifier for a problem can output 1 or 0.
In the case of f, a certificate would be x=1 or x=0 or similar.
Is this correct?
The problems are made of a few parts:

The problem, given x can we do y.
A certificate, an x.
A verifier, given y is it an actual answer for x.
For example, the compositeness problem, &quot;given an integer $X$, is it composite?&quot; might have a certificate $n_1, n_2$ where $X=n_1n_2$. This certificate would be checkable in time polynomial in the length of the certificate.