What is the application process, step by step?

Shortlist and read each bando.
Prepare documents: passport, degree certificate and transcript (or transcript so far), syllabus of each subject (M14), CV (often Europass), statement of purpose, recommendation letters, English certificate, portfolio for design or architecture.
Apply on each portal and pay any fee.
Interview or test, if the course has one.
Decision: admitted, conditionally admitted (M16), or rejected.
Pre enrol on Universitaly for the one course you will attend (M28).
Start CIMEA or the Declaration of Value for your bachelor's (M17).
Apply for scholarships (MAECI in spring, Invest Your Talent in spring, DSU in summer).
Apply for the visa once validated (M32).

Admission does not give you a right to a visa, and a visa does not give you a right to enrol (MUR circular).

This page explains official rules in plain language. I am not an immigration adviser and I do not give visa advice. For that, always use official government sources. Rules change: confirm with your university, the embassy and VFS before you act.