There are several different ways. You usually begin by getting a visa by the USCIS in the USA or at a USA embassy or consulate in your home country. The fastest way to obtain a Permanent Resident alien visa (green card) is through marriage to a US citizen or, more specifically, through a relative (a parent, spouse, or sibling over 21 years old who is a US citizen or a permanent resident alien who owns a green card). There are many different kinds of visas including employment, fiance/ee, and tourist (although tourist visas usually will never lead to a green card).
2 broad categories of green cards are as follows:
- Employment-based immigration which includes the EB1, EB2, and EB3 categories
- Family-based or marriage-based immigration
The applicable process depends on your current citizenship and family heritage. For the most up-to-date information contact your current home country's US Embassy.