If $M$ is a pseudo-Riemannian manifold and $\gamma\colon [a,b] \to M$ is a geodesic, then $\gamma(t_0)$ (with $t>a$) is said to be conjugate to $\gamma(a)$ along $\gamma$ if is there is a non-trivial Jacobi field $J$ along $\gamma$ such that $J_a = J_{t_0} = 0$.
I'd like to understand better the motivation behind this definition. I asked the professor in class and he said that conjugate points are related to critical points of the exponential map (and he did prove a result about that later), but I'd like to see a yet bigger picture: why should I care about conjugate points?
I apologize in advance in case this question is too soft.
As you said above, you can relate the above definition of conjugate points to critical points of the exponential map. Equivalently (use inverse function theorem), conjugate points are points in $T_pM$ where the exponential map fails to be a local diffeomorphism, or, equivalently, the images of these points under the exponential map.
This is important, since the set of all of these points (called the conjugate locus of $p$) will give us some information on how the exponential map fails to be a global diffeomorphism, which gives us information on geodesics. The other obstruction to global diffeomorphisms is the cut locus.
There are other equivalent ways of thinking about conjugate points. One that I like is that they are points where nearby geodesics starting at $p$ reconverge, which is actually the exact definition you get when you first start learning about them through Jacobi fields.
There is also an extremely important standard result you will probably encounter later on in your course which says that if a geodesic segment contains conjugate points then it cannot be minimal. All of this is done great justice in Milnor's classic book Morse Theory.