The existential quantifier doesn't mean "one", it means "at least one". So ∃x(student(x) ⋀ met(j,x)) translates as "John met at least one student". This formalization is consistent with structures in which there exist several students, but this should be ruled out, given that the English sentence uses the definite article "the".
"Structure" here means "context relevant to the discourse" -- of course, as Greg Lee points out, we don't claim that there exists only one student in the entire world, but the use of "the" suggests that there is only one student that comes into question as a relevant referent in the current discussion, and this situation is the one the formalization is supposed to adequately describe.
To assert that there is one unique student, we need to specify that there is no other individual which is also a student but different from that first introduced student: ¬∃y(student(y) ⋀ y ≠ x). You can easily verify that this can logically equivalently be expressed as ∀y(student(y) → y = x).
Whether the individual constant j means John or you depends on how you want to read it. A constant in a logical formula can't unambiguously refer to a concrete individual; its interpretation will depend on what we define. By convention, when translating to a formal representation we will usually choose meaningful names for non-logical symbols (= individual constants and predicates) and have an intended interpretation in mind; but on the logical side, there is nothing that prevents us from interpreting "j" as 2Lady Gaga" and "Student(x)" as "x is an elephant". Logic doesn't make predictions about the meaning of non-logical symbols; their interpretation is subject purely to convention. So to answer your second question, the formula could also mean "I met a student". But the use of the letter "j" makes it likely that John is indeed the intended interpretation.