OIS is overnight index swap: fixed float swap with floating rate based on some overnight rate
Traditionally (some examples):
- EONIA (EUR)
- Fed Funds (USD)
RFR: New Risk free rates (secured overnight funding rate):
In terms of what these curves look like: Reference is the underlying OIS. The curve uses instruments (Futures, Swaps) to construct (strip) discount factors and forward rates implied by those instruments. For example, a SOFR swaps curve will reference SOFR rate and use SOFR futures (not necessarily but possible, requires convexity adjustment though as future is linear in rates as opposed to convex FRA or swaps) and SOFR OIS swaps for varying maturities. These swaps for example are fixed - float with fixed leg typically having annual pay frequency and ACT/360 daycount. The float leg has daily reset frequency, annual pay frequency and ACT/360 daycount. ESTR is the same but usually using a basis spread to Euribor for longer tenors (at the moment) due to liquidity and market practice.
What you refer to is the current dual curve stripping framework. For example, swaption vol is now quoted with SOFR discounting, CME and LCH moved to SOFR PAI and discounting on Oct. 16 2020 on new AND legacy swaps.
For EUR cleared, major CCPs did this since July 27 2020.
The market switched to discounting with the relevant RFR rates on the dates above. Hence, if you have a dual stripped curve (e.g. 3m US libor), you use SOFR and no longer OIS (FF).
This really means that you discount the cashflows with OIS. Frequently, the OIS curve (SOFR for example) is stripped prior to Libor curve for example. So you find the present value of the fixed and float leg of a Libor swap, using OIS as discount factor. Thus, the only unknown values are the payments of the floating leg (PV fixed is already known). The exercise is to find discount factors of the floating leg that make both PVs equal.
It is not just discounting though (at least not looking forward). ISDA fallbacks will apply from 31 December 2021 for GBP, JPY, CHF and Euro-LIBOR and from 30 June 2023 for USD LIBOR. Even if there were some synthetic or "zombie" Libor after it officially ceases to exist, it is expected that liquidity will drop significantly. Note that the FED have issued supervisory guidance encouraging banks to “cease entering into new contracts that use USD LIBOR as a reference rate as soon as practicable and in any event by December 31, 2021”.
Once Libor is gone, your major reference will be RFR throughout.