In Europe, an ATCO license contains the following:
- One or more ratings
- One or more rating endorsements
- One or more unit endorsements
- One or more language proficiency endorsements
- A medical certificate
An ATCO certificate shall contain one or more of the following ratings:
- ADV - Aerodrome visual (tower control without use of radar)
- ADI - Aerodrome instrument (tower control with use of radar)
- APP - Approach procedural (approach control without use of radar)
- APS - Approach surveillance (approach control with use of radar)
- ACP - Area control procedural (area control without use of radar)
- ACS - Area control surveillance (area control with use of radar)
For each rating, the following rating endorsements are available:
ADV: None
ADI (at least one rating endorsement required):
- AIR (Air Control)
- GMC (Ground Movement Control) (without ground radar)
- TWR (Tower Control)
- GMS (Ground Movement Surveillance) (using ground radar)
- RAD (Aerodrome Radar Control)
APP: None
APS (may have one or more rating endorsements, but not required):
- PAR (Precision Approach Radar)
- SRA (Surveillance Radar Approach)
- TCL (Terminal Control)
ACP (no rating endorsements required):
ACS (can have one, but not two, rating endorsements):
- OCN (Oceanic Control)
- TCL (Terminal Control)
In addition, an ATCO will need a unit endorsement, which is the authorisation to provide air traffic control service for a specific sector, group of sectors and/or working positions under the responsibility of an air traffic service unit. Basically, a certification that you know the local regulations at the unit you are working at.
A language proficiency endorsement is also required in English and, if applicable, in the local language if required.
Furthermore, there are instructor ratings available, but I think that it beyond the scope of your question.
Then of course all controllers need to hold a class 3 medical certificate for their ATCO license to be valid.
As you can see, with the number of ratings and rating endorsements available, there are a lot of different possible combinations. So the answer to "what types of air traffic controllers are there?" is not a simple one. Regardless, I hope the above gives you a general overview of how ATC licensing is structured in Europe.