Adding a new column is one of the most common schema changes. It can be simple, but under load it can also take down an application. The impact depends on the database engine, table size, indexing, and the way the migration is executed.
In PostgreSQL, ALTER TABLE ADD COLUMN with a default value before version 11 rewrites the table. On large datasets, that rewrite can lock writes and bloat IO for minutes or hours. The safer approach is to add the new column without a default, then backfill values in small batches, and finally set the default and NOT NULL constraint after the backfill.
In MySQL, ALTER TABLE often copies the table. This can explode downtime unless you use ONLINE DDL (InnoDB) or tools like pt-online-schema-change. Even then, watch for triggers, foreign keys, and replication lag.