The E.I.T. isn't a thing, it's a title.
An Engineer-In-Training (E.I.T.) is the title that California and some other states bestow on an engineer that has passed the Fundamentals of Engineering (F.E.) exam, administered by the National Council of Examiners for Engineering and Surveying (NCEES). Other states use the title Engineering Intern (E.I.)
Often times, the F.E. exam is referred to as the E.I.T. exam. The F.E. exam is administered in April and October and students take it as early as their junior year, but usually in their senior year. It can be taken after graduation.
The exam is 8 hours, multiple choice. The first 4 hours are a general exam, taken by all engineering majors, the last 4 hours can be either a general or specific exam (chemical, civil, electrical, environmental, industrial, or mechanical). The exam covers chemistry, computers, dynamics, electric circuits, engineering economics, ethics, fluid mechanics, materials science, mathematics, mechanics of materials, statics, and thermodynamics.