What is the minimum distance allowed between planes in flight, in Europe?
Separation minima in Europe are pretty much identical to the rest of the world, since almost all countries base their aviation regulations on ICAO recommendations.

To summarize:

The minimum vertical separation is 1000 ft. In theory, above FL290, the minimum is 2000 ft, but since all European countries have implemented reduced vertical separation minima (RVSM), the 1000 ft. rule applies all the way up to FL410.

When horizontal separation is based on surveillance (such as radar), the minimum separation generally is 5 nautical miles (NM). Around larger airports with excellent radar coverage and at lower altitudes (where planes tend to fly slower), the minimum is routinely reduced to 3 NM. On final approach path to some runways, it can be reduced further, typically to 2.5 NM. They can get even closer in case of parallel runway operations, but that's a bit too complex to cover here. Special rules apply to mitigate the risk of wake turbulence. For example, the separation between a light aircraft flying behind a medium aircraft can never be reduced below 5 NM.

Horizontal separation based on position reports (no radar coverage) is pretty rare, but still in use in some areas. Here, the minimum horizontal separation is typically not defined as a distance, but instead can be based on a navigation aid (for example, two aircraft on VOR radials diverging by at least 15 degrees, and one of them is at least 15 NM from the VOR, are separated). Another option is lateral separation, which is basically when aircraft operating in different areas are defined to be separated, regardless of the actual distance between them (for example, an aircraft flying along airway A is separated from an aircraft flying along airway B, since airway A and B never cross). Rules for lateral separation are defined locally.

Finally, in the vicinity of aerodromes, separation can be reduced further (often referred to as "visual separation"). Essentially, if the tower controller has two aircraft in sight, and can determine visually that there is no risk of collision, they are separated (regardless of the actual distance between them).

Remember that all the separation minima above only apply in areas where ATC is required to provide separation. If flying in class D airspace, for example, (used in many control zones and terminal areas) ATC is not required to separate VFR traffic from IFR traffic. Instead, ATC will pass traffic information, allowing the pilots to visually recognise other traffic on their path, and take appropriate action to avoid a risk of collision in accordance with the rules of the air.