Let's make some definitions more clear before we move on to the intuition, so that we can be sure we are talking about the same things. In this context, a path is some possibly curved line between two endpoints. A parameterization of a path are the formulas $x = x(t)$ and $y = y(t)$. You can think of a path as some geometric object and a parameterization as some equations describing the path. The key here is that a path can have several parameterizations; for example, the straight line from $(0, 0)$ to $(1, 1)$ can be parameterized as $x = t, y = t$ or $x=t^2, y=t^2$, for $t \in [0, 1]$.
Now what about line integrals? A line integral is an integral over a path; ordinarily we compute a line integral using a parameterization of a path, but the original line integral won't make mention to a specific parameterization of said path. This is because any parameterization of a path will yield the same line integral. You can try this out with any line integral you will come across right now by using different parameterizations. Indeed, some parameterizations will be easier to calculate the integral than others.
Let's contrast this with what your question is asking. There are some line integrals where only the endpoints of the path of integration actually matter; the path itself can be anything, so long as the endpoints are as specified. This is much different than have different parameterizations (we have whole different paths!). These line integral for which the path of integration "doesn't matter" are called conservative, and I'm sure you will cover these types of fields in your course.
As a classic example, think of gravity (say, at the surface of the earth). We know an object falls from point $A$ to point $B$, and the question is how much work gravity has done to this object. Since gravity is a constant unidirectional force, you can convince yourself that no matter what meandering path is taken, gravity will do the same amount of work. For example, if the object goes up, the object must come back down, and so that initial negative work done by gravity is counteracted by the positive work when the object comes back down. Alternatively, if you have taken a physics course, you know that we can calculate the work done by gravity in this example using potential energy. This concept of a potential function is critical to conservative fields.
As a classic non-example, take friction. The longer and more winding the path you take, the more frictive forces act on the object, and the more work is done. Thus path matters here when calculating work (but again, parameterization does not).