TDi is Turbo Diesel injection and HDi is High-pressure Direct injection.

They're both terms to describe modern turbodiesel engines.

TDI stands for either Turbo Diesel Intercooled, or Turbo (Diesel) Direct
Injection.  Some people may argue what it really stands for, but personally
I'd say the latter, as I think there might be non-intercooled TDI engines,
that are obviously still direct injection.  It's generally just used by VAG
(Volkswagen Audi Group), who make, other than the obvious, Seat, Skoda and
Bentley (though I don't think they've done a diesel Bentley yet, but it
probably isn't far off).

(You're gonna want to know what Intercooled and direct injection mean next,
aren't you? :-))

HDI is a common-rail turbo diesel engine made by Peugeot/Citroen.  It's in
theory a bit more fancy than VAG's TDI engines, due to fuel running at a
much higher pressure, giving (in theory) better performance and economy.
But then VAG use a similar system called Pump Deuse (PD) - well maybe not
that similar, but same sort of results, and also some of their latest V6 TDI
engines actually use commonrail technology, but they didn't change the name,
'cos they didn't need to, as people know their engines are bloody good
anyway.