1st = 1 point
2nd = 2 points
3rd = 3 points
>75% of the race completed = 4 points
50-75% completed = 5 points
25-50% completed = 6 points
<25% completed = 7 points
Not entering, not starting or disqualification from the race = 8 points
In the case of shared drives only the original driver who started the race in the shared car would score the points (e.g. at the 1951 French GP Fagioli would have been awarded the 1 point for winning while Fangio would have scored 4 points for his original car that he handed over to Fagioli).
List of champions
1950: Juan Manuel Fangio (Alfa Romeo)
1951: Juan Manuel Fangio (Alfa Romeo)
1952: Alberto Ascari (Ferrari)
1953: Alberto Ascari (Ferrari)
1954: Juan Manuel Fangio (Maserati & Mercedes)
1955: Juan Manuel Fangio (Mercedes)
1956: Peter Collins (Ferrari)
1957: Juan Manuel Fangio (Maserati)
1958: Drivers: Mike Hawthorn (Ferrari) Constructors: Ferrari
1959: Drivers: Jack Brabham (Cooper-Climax) Constructors: Cooper-Climax
1960: Drivers: Jack Brabham (Cooper-Climax) Constructors: Cooper-Climax
1961: Drivers: Phil Hill (Ferrari) Constructors: Ferrari
1962: Drivers: Graham Hill (BRM) Constructors: BRM
1963: Drivers: Jim Clark (Lotus-Climax) Constructors: Lotus-Climax
1964: Drivers: Graham Hill (BRM) Constructors: BRM
1965: Drivers: Graham Hill (BRM) Constructors: BRM
1966: Drivers: Jack Brabham (Brabham-Repco) Constructors: Brabham-Repco
1967: Drivers: Denny Hulme (Brabham-Repco) Constructors: Brabham-Repco
1968: Drivers: Graham Hill (Lotus-Ford) Constructors: Lotus-Ford
1969: Drivers: Jackie Stewart (Matra-Ford) Constructors: Matra-Ford
1970: Drivers: Jack Brabham (Brabham-Ford) Constructors: Lotus-Ford
1971: Drivers: Jackie Stewart (Tyrrell-Ford) Constructors: Tyrrell-Ford
1972: Drivers: Emerson Fittipaldi (Lotus-Ford) Constructors: Lotus-Ford
1973: Drivers: Jackie Stewart (Tyrrell-Ford) Constructors: Lotus-Ford
1974: Drivers: Clay Regazzoni (Ferrari) Constructors: McLaren-Ford
1975: Drivers: Niki Lauda (Ferrari) Constructors: Ferrari
1976: Drivers: James Hunt (McLaren-Ford) Constructors: Ferrari
1977: Drivers: Jody Scheckter (Wolf-Ford) Constructors: Ferrari
1978: Drivers: Mario Andretti (Lotus-Ford) Constructors: Lotus-Ford
1979: Drivers: Gilles Villeneuve (Ferrari) Constructors: Ferrari
1980: Drivers: Alan Jones (Williams-Ford) Constructors: Williams-Ford
1981: Drivers: Carlos Reutemann (Williams-Ford) Constructors: Williams-Ford
1982: Drivers: Keke Rosberg (Williams-Ford) Constructors: McLaren-Ford
1983: Drivers: Alain Prost (Renault) Constructors: Brabham-BMW
1984: Drivers: Niki Lauda (McLaren-TAG) Constructors: McLaren-TAG
1985: Drivers: Alain Prost (McLaren-TAG) Constructors: McLaren-TAG
1986: Drivers: Nelson Piquet (Williams-Honda) Constructors: Williams-Honda
1987: Drivers: Nelson Piquet (Williams-Honda) Constructors: Williams-Honda Jim Clark Trophy: Jonathan Palmer (Tyrrell-Ford) Colin Chapman Trophy: Tyrrell-Ford
1988: Drivers: Alain Prost (McLaren-Honda) Constructors: McLaren-Honda
1989: Drivers: Alain Prost (McLaren-Honda) Constructors: McLaren-Honda
1990: Drivers: Ayrton Senna (McLaren-Honda) Constructors: McLaren-Honda
1991: Drivers: Ayrton Senna (McLaren-Honda) Constructors: McLaren-Honda
1992: Drivers: Nigel Mansell (Williams-Renault) Constructors: Williams-Renault
1993: Drivers: Alain Prost (Williams-Renault) Constructors: Williams-Renault
1994: Drivers: Damon Hill (Williams-Renault) Constructors: Williams-Renault
1995: Drivers: Michael Schumacher (Benetton-Renault) Constructors: Benetton-Renault
1996: Drivers: Jacques Villeneuve (Williams-Renault) Constructors: Williams-Renault
1997: Drivers: Jacques Villeneuve (Williams-Renault) Constructors: Williams-Renault
1998: Drivers: Mika Häkkinen (McLaren-Mercedes) Constructors: McLaren-Mercedes
1999: Drivers: Eddie Irvine (Ferrari) Constructors: Ferrari
2000: Drivers: Michael Schumacher (Ferrari) Constructors: Ferrari
2001: Drivers: Michael Schumacher (Ferrari) Constructors: Ferrari
2002: Drivers: Michael Schumacher (Ferrari) Constructors: Ferrari
2003: Drivers: Michael Schumacher (Ferrari) Constructors: Ferrari
2004: Drivers: Michael Schumacher (Ferrari) Constructors: Ferrari
2005: Drivers: Fernando Alonso (Renault) Constructors: McLaren-Mercedes
2006: Drivers: Fernando Alonso (Renault) Constructors: Ferrari
2007: Drivers: Lewis Hamilton (McLaren-Mercedes) Constructors: Ferrari
2008: Drivers: Felipe Massa (Ferrari) Constructors: Ferrari
2009: Drivers: Jenson Button (Brawn-Mercedes) Constructors: Brawn-Mercedes
2010: Drivers: Fernando Alonso (Ferrari) Constructors: Red Bull-Renault
2011: Drivers: Sebastian Vettel (Red Bull-Renault) Constructors: Red Bull-Renault
2012: Drivers: Sebastian Vettel (Red Bull-Renault) Constructors: Red Bull-Renault
2013: Drivers: Sebastian Vettel (Red Bull-Renault) Constructors: Red Bull-Renault
2014: Drivers: Lewis Hamilton (Mercedes) Constructors: Mercedes
2015: Drivers: Lewis Hamilton (Mercedes) Constructors: Mercedes
2016: Drivers: Lewis Hamilton (Mercedes) Constructors: Mercedes
2017: Drivers: Lewis Hamilton (Mercedes) Constructors: Mercedes
2018: Drivers: Lewis Hamilton (Mercedes) Constructors: Mercedes
2019: Drivers: Lewis Hamilton (Mercedes) Constructors: Mercedes
So let's begin with the inaugural season:
1950
A very interesting one to start with. Fangio's reliability problems during the season don't impact his chances as much as in reality because he got far enough into the races he retired from to minimise the damage. Farina's chances on the other hand were handicapped by the first lap pile-up at Monaco and scoring 7 points as a result. Going into the final race Fangio has a 4 point lead over Fagioli with Farina a further 2 points back. Fangio's gearbox failed just past 25% distance, meaning he would score 6 points, putting Farina out of contention. Fagioli had to win to take the title but in the end only finished 3rd.
Because so many drivers take part and score points I will only list the top 20 for most seasons. Also note how the Indy 500 doesn't interfere with the standings as much.
Final standings:

1. Juan Manuel Fangio 25 pts.
2. Giuseppe Farina 26 pts.
3. Luigi Fagioli 26 pts.
4. Louis Rosier 36 pts.
5. Philippe Étancelin 37 pts.
6. Johnny Claes 37 pts.

7. Alberto Ascari 43 pts.
8. Louis Chiron 43 pts.
9. Raymond Sommer 43 pts.
10. Prince Bira 44 pts.
11. Toulo de Graffenried 44 pts.
12. Yves Giraud-Cabantous 46 pts.
13. Pierre Levegh 47 pts.
14. Bob Gerard 48 pts.
15. Luigi Villoresi 48 pts.
16. Cuth Harrison 48 pts.
17. Toni Branca 48 pts.
18. Johnnie Parsons 49 pts.
19. Geoffrey Crossley 49 pts.
20. Bill Holland 50 pts.