What kinds of engineers are there and how do they differ?

There are many different kinds of engineers but the major ones are electrical, mechanical, civil, chemical, biomedical, and computer.

All engineers study similar things such as math and science. however depending on the major there are some differences in specializations
Electrical engineers take engineering and focus more on electricity and magnetism, signals (think about your remote control and cell phone works). Electrical engineers in general work on electrical systems. It could be small like a circuit board in consumer electronics. It could also be big like engineering a blueprint to wire a 100 story office building.

Mechanical engineers focus on physics, kinematics and thermodynamics. So working on things similar to the classic physics problem of throwing a ball into the air and determining it's velocity, acceleration etc. So mechanical engineers can work on rockets and determine aerodynamics, thrust to weigh ratios, structural integrity of the materials used, etc. Any physical thing that has some force applied to it and is engineered falls into the space of mech eng. airplanes, cars, etc.

Civil engineers also focus on physics similar to mechanical engineers, however civil engineers focus more on civics projects like designing bridges, making sure that they can handle all the weigh that they hold year in and year out. In the real world budget concerns come into play so you can't just throw all the steel into a bridge. it has to be calculated to be efficient.

So in the same pattern, chemical engineers focus on chemistry and use engineering in that field.
biomedical engineers use biology and engineer to create for example x-rays, MRI machines, etc.
Computer engineers focus on computer science and electronics and magnetics to make powerful, small, cheap computers.

Think about what science you like most and what schools offer that program. The math will be pretty similar throughout each major.

Engineers design so anything that can be designed has an engineering application. Engineers also test the designs to verify sound engineering practices were carried through design to construction. Scientist explore and search for answers to questions. Most fields have an engineering and scientific aspect. Scientist collect data to formulate the rules within which engineers work. Both can be either experimental (hands-on, in-the-field) or theoretical e.g., use software models to collect data.