I worked as a PADI scuba instructor in the Cayman Islands for several years.
There are three levels of options: Discover Scuba Diving, Scuba Diver, and Open Water Diver.
**Discover Scuba Diving **consists of a brief classroom session and one pool session. A dive at a shallow site (40ft or 12m max), such as a reef, accompanied by a dive professional may, and usually does, follow. This can all be done in one day. You may be able to arrange more such shallow dives in subsequent days.
Discover Scuba Diving is not a certification course. You would need to repeat the entire course if you want to dive again after a year has passed since the initial course. And it is always the dive professional’s discretion if they are willing to offer additional shallow open water dives after the DSD course. The pool and open water sessions of a Discover Scuba Diving course might can be applied, at the instructor’s discretion, to a Scuba Diver or Open Water Diver certification course.
Open Water Diver is the typical entry level certification course. It involves covering the material in five chapters of a textbook, end of chapter written reviews, quizzes, and a final exam. There are five sessions in confined water, typically a swimming pool, to learn and practice various skills. And there are at least four open water certification dives in which you must demonstrate mastery of the skills.
The full Open Water Diver course can be completed in as little as 3 days, though you will need to read the text and complete the end of chapter reviews outside that time frame. Three chapters in the classroom on morning one, and three sessions in the pool in the afternoon. The remaining two chapters in the classroom on morning two, and two open water dives on afternoon two. Two pool sessions on morning three, and two remaining open water dives on afternoon three. It is nice to have an extra half day and not fee so rushed.
Upon completing the Open Water Diver course you are certified to conduct dives in conditions as good or better as what you experienced in your training course to a maximum depth of 60ft (18m) while accompanied by any other similarly certified diver.
Scuba Diver is a certification which is a subset of the Open Water Diver course. It consists of only the first three chapters of classroom material, three pool sessions, and two open water certification dives.
Upon completing the Scuba Diver certification you may dive to a maximum depth of 40ft (12m) while accompanied by dive professional (divemaster or instructor). This essentially is like a permanent DSD course, where you could do these accompanied dives without repeating the classroom and pool session of a DSD each time you go on vacation. A Scuba Diver course can be completed in two days, if you read the textbook and do the end of chapter written reviews outside of class time.
Many scuba tour operators in dive destinations organize their dives assuming a minimum Open Water Diver certification. A Scuba Diver certification may not be sufficient to go out on the typical two tank morning dive trip.
Whether for a Scuba Diver or Open Water Dive certification it is possible to do some of the work before a trip to a dive destination. You can complete the classroom material online at your leisure before you trip. If you wish you may also complete the swimming pool training dives as well. You instructor then fills out paperwork for a referral and you complete the open water certification dives at your destination over at least two days.
**TL;DR version: ** 2 or 3 days, depending on level of certification.