OTF (Open Type Font) are cross-platform (Mac and PC) fonts. They typically have a smaller file size which allows for a quicker download. They support expanded characters which allows for alternate characters and ligatures.
TTF (True Type Font) are similar to OTF fonts but they require separate files for each instance of the font. For example, a separate .ttf file is needed for bold, italics, or bold italics.
For most intent and purposes, it will make little difference to use one or the other file type when downloading fonts from LD unless you are using one of our "Smart Fonts". A "Smart Font" is a font with alternate characters and ligatures. 
We will always generate a .ttf of a "Smart Font" but you will not be able to have the font automatically replace specific letter combinations when you type them unless you are using a .OTF font and a program that supports the ligature features of a .OTF font. Adobe programs have always supported these features. Other programs have been slower to come on board but are doing so daily. Please use a search engine to make sure the programs you use support those features. If not there are work arounds.
To use ligatures in OTF fonts in Windows without supporting programs like Adobe, see this tutorial.
To use ligatures in OTF fonts on a Mac without supporting programs like Adobe, see this tutorial.