In this class, a certificate is defined to be a string which can be used
to verify that a certain other string is in a certain language.

In human terms, your passport certifies that you are a citizen of the
United States, or some other country.  If you don't have your passport
with you, it may still be possible to determine that you are a citizen,
but the process could take much longer.

The general format of certificates is this.  Suppose L is a language.
A certifier for L is defined to be a machine M such that, given any
string w, w is in L if and only if there is another string c (called
a certificate for w) such that the string w#c is accepted by M.
We assume that '#' is not in the alphabet of L.

We would like M to execute quickly.

A language L is in the class NP if and only if it has a certifier M such
that every w in L has a certificate c such that M accepts w#c in time which
is polynomial in the length of w.

For example, boolean satisfiability is the class NP.  The string
 (x1+~x2+x3)*(x2+x3)*(~x1+x2)*(~x1+~x2+~x3)
has the certificate
 x1=1;x2=1;x3=0;

Despite the fact that boolean satisfiability cannot be decided in polynomial
time by any known algorithm, you could easily write a certifier for boolean
satisfiablility that executes in polynomial time.