Introduction to the Basic-Concepts of Bose-Einstein Condensation

This text corresponds to part of the course presented at the ELAF XXXVIII. It is composed of work previously done along the years. The level of this introduction follows the last year of undergraduate and first year of graduate courses. We apologize to the experts on this field, but the idea here is to provide the basic principles and tools for students just getting involved with this topic. For those interested in a deeper reading on the subject, we strongly recommend the review article published by Courteille, Bagnato and Yukalov [1] from which part of this text was extracted. The lecture is divided in three parts: an introduction, the basic concepts of Bose-Einstein Condensation and information about making and probing BECs.