The blog post by Charlie Custer explains the use of the SQL command ADD CONSTRAINT, which is employed with ALTER TABLE to introduce constraints like primary keys, foreign keys, UNIQUE, and CHECK constraints to existing tables in a SQL database. Constraints in SQL are crucial for maintaining data integrity by allowing only data that meets predefined rules to be entered into tables, thereby preventing errors and ensuring proper data relationships. The article provides examples of adding various types of constraints, such as primary key constraints to uniquely identify rows, foreign key constraints to link data across tables, UNIQUE constraints to ensure distinct values in a column, and CHECK constraints to validate data against a Boolean expression. It also covers how to drop and rename constraints using DROP CONSTRAINT and RENAME CONSTRAINT commands, respectively, and emphasizes the importance of managing schema changes without causing database downtime, a feature supported by CockroachDB.