Module and diametral pitch are terms to describe the size and spacing of gear teeth, with dp usually describing gears of imperial measurement and module metric, though is entirely possible to express a module as dp or dp as module.
Pitches for threads, which is what you're going to be using a lathe for, are described using the number of teeth per inch - similar to diametral pitch for gear teeth - for imperial threads, with metric being specified by how much one turn advances the thread, or "lead".
With a change gear lathe, it's possible to make any thread you can imagine regardless of the leadscrew pitch, providing you have or are prepared to make the required gears. With a thread cutting gearbox, you often have some options for cutting threads of the type that your leadscrew isn't, though metrics on a lathe with an imperial leadscrew are often approximations only, and vice versa. The smallest gear that can correctly convert inches to mm is 127 teeth - quite a thing to fit in a small lathe. There are some smaller gear combinations that result in a ratio close to 1.27:1, and these are often used, but 127 being a prime, nothing spot on. Mostly the inaccuracy is academic for hobby purposes.
We're in odd positions either side of the pond with standards and what equipment is available. Here, most work is metric, though most old machines available are imperial. I'm so glad my mill is metric - I'm no great fan of the (sorry) imperial system - but my lathe is imperial and that needs to change as I work in metric. In the US, you're still using inches in a largely metric world, but most machinery is still set up for inches. I guess it depends what you want to do?