What is ADS-B and who needs it?
To answer your first question, ADS-B stands for Automatic Dependent Surveillance - Broadcast. 


Automatic in the sense that it operates without external stimulus (unlike radar)
Dependent in the sense that the surveillance information is derived from on board systems
Surveillance in the sense that it's primary intended to provide surveillance information to other parties
Broadcast because it transmits to all listeners without knowing who those listeners are, there is no two-way communication.


The capability of an aircraft to transmit ADS-B signals is called ADS-B OUT, the capability of an aircraft to receive ADS-B signals is called ADS-B IN.

ADS-B data contains information about the identity, position, altitude and velocity of the aircraft. The position and velocity information is derived from GNSS (GPS). 

There are three technologies defined for ADS-B 


1090ES is operating on 1090 MHz, the same frequency used for Secondary Surveillance Radar (SSR) and Airborne Collision Avoidance Systems (ACAS/TCAS). Messages are transmitted by the same transponder that also replies to radar interrogations. This will be a Mode-S capable transponder because the data is formatted in a similar way. Since this frequency is the worldwide used standard for aviation, international aviation will use this ADS-B standard.
UAT (Universal Access Transceiver) works on 978 MHz. A special transceiver is required to use this form of ADS-B. The transceiver is able to send and receive data, it will also receive ADS-B transmissions from other aircraft. UAT is only allowed within the USA and China. The frequency 978 MHz is reserved for other purposes (DME/TACAN) in other countries.  To encourage people to equip with ADS-B voluntarily, the FAA provides additional services on UAT. These are Traffic Information Service - Broadcast (TIS-B) which transmits position of aircraft detected by FAA's SSR's and Flight Information Service - Broadcast (FIS-B) which provides the receiver with weather information and operational data such a airspace restrictions and NOTAMS. 
VHF Datalink Mode 4 (VLDm4) uses a frequency in the 118-136.975 MHz block, which is internationally reserved for aviation. Normally these frequencies are used for voice communications, so a special radio is required to support VDLm4. Just like UAT, VDL mode 4 allows for services like TIS-B and FIS-B. VDLm4 was tried in Sweden and Russia, but has not been able to gather much mass. It is unlikely that it will be used on a large scale.




Since you mention 2020 as the year by which aircraft should be equipped with ADS-B I am going to assume you are from FAA-land. 

The FAA mandates aircraft in the 48 contiguous states (and D.C.) that operate in class A, B, or C airspace or class E airspace above 10000ft (class E airspace below 2500 AGL excluded) to be ADS-B OUT equipped by January 1st, 2020. 
The mandate itself can be found here and the associated Advisory Circular (AC90-114) can be found here. 

Aircraft operating below 18000 ft can use either UAT or 1090ES. Above 18000ft, 1090ES is to only approved ADS-B technology. 



Countries that have ADS-B mandates are (this list is probably not exhaustive)


Australia
Canada
(Hudson bay)
China
Countries in the European Union
Fiji
Hong Kong
India
Indonesia
Seychelles
Singapore
USA
Vietnam