What is CBIR & Basic IR? I know that the IR - Instrument rating - is the full-blown option to be allowed to fly IFR. I also know that EIR - En-route Instrument rating - is an option to have a license that's restricted to fly IFR except for during takeoff and landing.

What is CBIR and what is Basic IR? I also heard that Basic IR will "come soon" and "replace EIR". What's the roadmap for EASA instrument ratings in the next years?
The Competency Based Instrument Rating (CBIR) allows you to gain an IR in less hours than the "full" course, by either crediting previous experience or licenses.

I held a UK issued IR(R) (old IMC rating), which meant that I had approx 60 hours IFR time when starting my IR. A full IR would have meant 45 hours training, whereas the CBIR allowed me to do just 15 hours (in accordance with EASA Part FCL, Appendix 6). 

The BIR, as far as I am aware, is effectively going to mirror the UK IR(R) but make it a full EASA rating. That isn't confirmed however, as I haven't yet seen any official documentation from EASA.