While studying the Solar System, I found that some galaxies are either elliptical or elongated. What's the difference?
tl;dr: "Elliptical" refers to a special kind of galaxies. "Elongated" is a loose term meaning that a galaxy is stretched out in one direction.
Etymologically, "elliptical" means having a shape that can be described by a certain mathematical function — the ellipse. For a three-dimensional object, the proper term is "ellipsoidal", but still it refers to being described by a certain equation. In contrast, "elongated" is a more loose term, meaning something like "significantly deviating from a spherical shape in one direction".
In the context of galaxies, the term "elliptical" refers to a certain type of galaxies that tends to be not actively star-forming, have no or very little structure (as opposed to the beautiful spiral arms of spiral galaxies), be quite massive, as well as other characteristica. Having no ongoing star formation, the massive, short-lived stars, which have blue colors, have died long ago, leaving behind the less massive, red stars. In addition, elliptical are often quite dusty, further reddening the light. Thus, ellipticals appear red in color.
The shape of an elliptical galaxy is described by the relative size of its three axes, $a$, $b$, and $c$. If two axes, say $a$ and $b$, are roughly the same size and are larger than $c$, the galaxy is said to be "oblate", whereas if they are less than $c$, it is said to be "prolate".
As in the etymological meaning, an elongated galaxy is not a specific type, but rather refers to a galaxy departing from spherical shape in one direction. This could for instance be a prolate elliptical, but it could also by any other galaxy that has been distorted by merging with another galaxy.
Usually, though (I think), it will refer to ellipticals. The "elongation" of an elliptical is defined as $10\times(1 - b/a)$, where $a$ and $b$ is now the observed axes, i.e. of the two-dimensional projection on the sky. Ellipticals are classified according to this number as E0 (being spherical, such that $a=b$) to E7 (being very elongated). In principle, you could have even higher E-numbers, but that's not observed.
Elongated means made longer. An elongated shape or object is stretched primarily in one direction.
Elliptical means having an ellipse-like shape. An ellipse is an elongated circle, where the elongation is done in a constant/linear way.
For example, a limousine often has the form of an ordinary car, elongated (an elongated car). It is not at all ellipse-like, however.
More simply, "elongated" is a general term that can be used to describe the appearance of any galaxy, and means more or less what it means in ordinary English.
"Elliptical" is a specific technical term for galaxies, and refers to galaxies which are in simplistic terms "round and fuzzy" -- made up primarily of stars on randomized three-dimensional orbits, with little or no gas and little or no current star formation, without a dominant stellar disk. They can be "elongated", but they can also be (intrinsically, or when seen at the right orientation) circular. There's a decent introductory Wikipedia article.