What do the terms MDI and SDI mean in relation to the software interface?

What are they?

Names already say it all. MDI (Multiple Document Interface) translates to multiple document Interface because it is a choice of interaction method in which an application allows the user to manipulate multiple "documents" in the same session (the same instance of the application). On the other hand, SDI (Single Document Interface ) translates to single document Interface because it deals with the choice of interaction method where the user only you can handle a single "document" at a time (or for each instance of the application).

Classic example are text editors. Notepad, traditional Windows, for example, only allows you to edit a file:

On the other hand, Notepad++ already allows you to edit several of them at the same time by selecting the asset from tabs (one for each open file):

Note that I previously used the word "document" in quotation marks. The reason is that such a "document" can be anything other than a file with text. A music editing application using an SDI approach would allow you to edit only one song at a time, while another using the MDI approach would allow you to edit several, probably switching between them via tabs or internal windows. In fact, there are several design standards for MDI applications, and tabs are just one of them. In the old days, used internal mini-windows in the "parent" window (the main one of the application):

But today it is more common to use "dockable" tabs or windows (which can be "docked" in pre-established regions or moved to "float" as independent Windows).

What are they for?

When a designer considers using one or another approach to interaction, he considers the use that users will make of such "documents". The main question is:

Will the user need to handle more than one file simultaneously?

If the answer is yes, the option for an MDI approach may be the most appropriate. I say Can because nothing prevents you from allowing users to open more than one instance of the same application and edit each "document" in a separate instance. In this case, an important additional question is:

What will be the interactions between "documents" that the user need do?

Because if the user needs to interact between documents quite often (for example, look at something that is in one to decide how to change the other, or copy and paste, etc), it can make much more sense from the point of view of usability you provide an MDI interface.