Mechanical Engineering (ME) involves designing and manufacturing everything from small individual parts and devices (e.g., microscale sensors and inkjet printer nozzles) to large systems (e.g., spacecraft and machine tools). [1]

The role of a mechanical engineer is to take a product from an idea to the marketplace. In order to accomplish this, a broad range of skills are needed including mathematics, physics, material science, thermal and mechanics, and manufacturing.  the frontier of research includes Robotics, Micro Electro-mechanical Systems (MEMS), Mechatronics, Nanotechnology, Biomechanics, Acoustical engineering and etc. The industry includes health, environment, energy, security and electrical engineering. 

The top  university offering ME are [4]:  

1. Massachusetts Institute of Technology
2. Georgia Institute of Technology
3. University of California -Berkeley
4. Standford
5. University of Michigan--Ann Arbor
6. Purdue University--West Lafayette
7. University of Illinois--Urbana-Champaign
8. Cornell University
9. California Institute of Technology
10. University of Texas
11. Carnegie Mellon University
12. Princeton 

To complete undergraduate, an example course list can be found at MIT - Engineering, Mechanical Engineering (Course 2- A).

Let's look at some ME projects[3]:

• MIT - Harvesting Fresh Water from Fog (video)
• MIT - Engineering Fluids at the Nanoscale (video)
• MIT - Introduction to Robotics at MIT (video)

Reference 

1. What is Mechanical Engineering? by Columbia Engineering 
2. Mechanical Engineering at Wikipedia
3. MIT- MECHE VIDEO LIBRARY
4. Mechanical Rankings by US News​