“New York-bound local train, now boarding on track three.”

You're waiting at a train station for a train to New York. You hear an announcement which says:

New York-bound local train, now boarding on track three.

a (somewhere)-bound (bus/train/airplane)

This phrase expresses the direction that something is travelling in:

There's a Hong Kong-bound flight leaving in an hour and a half.

No, don't take the south-bound train.

You can also say "bound for ___":

Is this bus bound for New Jersey?

a local train

A "local train" stops at every stop on its line.

Some trains, called "express trains", skip some stops. They do this to get to the destination faster.

(a train is) now boarding

When a train, bus, or airplane is "boarding", it means that passengers are being let onto it.

When airport or train station employees are making announcements, they say:

Now boarding.

But when you want to talk about the flight or train that you're taking, you say:

My flight's boarding now, so I've got to go. I'll call you later.

I think our train is boarding.

track three

Different train tracks in a train station are usually identified with different letters or numbers.