It is not that complicated. A total of 32 teams will compete in the UEFA Champions League. Which association will send how many teams is decided by the UEFA coefficients. Based on these numbers, associations like Spain, England and Germany could send four teams each and associations like Romania, Cyprus and Denmark, etc. could send only one team (based on the 2015-16 competition), those teams from associations with low coefficients should participate in First Qualifying Rounds, Second Qualifying Rounds, Third Qualifying Rounds.
After these rounds, a total of 22 teams:
12 champions from associations 1 to 12
6 runners-up from association 1 to 6
3 third-placed teams from association 1-3
Europa League title holders
10 winners from the third qualifying round will make 32 teams.
Therefore, the maximum number of teams that each association could send is only 5 when a title holder is not from the top four teams in the league.
There are four pots for drawing for each team.
Pot 1: Seven champions from the top seven countries + a title holder (winner of the last year's UEFA Champions League). If the title holder is from the 12 champions, there will be eight champions from the eight top countries.
Pot 2: Remaining five champions that didn't make it to Pot 1 + three runner-ups
Pot 3: Remaining three runner-ups + three third-placed teams + two qualifying round winners.
Pot 4: Eight remaining qualifying round winners.
There is another pot that has balls containing group names.
Drawing procedures are quite simple. For example, one ball will be drawn from Pot 1 (champions) and another ball will be drawn from the pot for group A to H to determine the group for the team. It is repeated until all the balls in Pot 4 are drawn. A group (which already has a team from the same country) can't be drawn for a team, because the ball contains the group will be eliminated before a drawing to ensure no team can play a club from their own association. For example, after ManU is drawn into Group A and Chelsea is drawn later, a ball that has Group A will not be in the pot.
A side note: For the 2015-16 Champions League, UEFA decided not to draw Russia and Ukraine into the same group based on the political situation between the two countries.