There are three main types of tournaments that teams will play in: national leagues, national knockout cups, and continental-level competitions which are (mostly) knockout.
National leagues - these are the competitions in which a team will play most of its games. Pretty much every country in Europe will have its own - think the English Premier League, the Spanish Primera Division (which is nicknamed 'La Liga' by most people - they're the same competition), the German Bundesliga, the French Ligue 1, etc. These divisions normally have somewhere between 14 and 20 teams, with each team playing each other team twice during the course of a regular season (once home and once away) - so in the English Premier League for example, that's 38 games (one home and one away against each of the other 19 teams). There's no playoff system - whichever team does the best over those 38 games wins.
The vast majority of European leagues, and certainly all the major ones, will have a system of promotion and relegation - i.e. the bottom few teams will drop down at the end of the season and play in the second-tier division the next year, while the best few teams from that division will come up and play in the top division. In some countries this system extends through huge numbers of leagues - I believe in England the league 'pyramid' covers more than 20 levels, so a team that's currently right at the bottom of level 20 can theoretically, given enough time, ascend right to the top of the top division.
National cups - these again only involve teams playing against teams from their own country, but this time all those teams playing in the lower divisions get involved as well. It's a straight knockout - all the teams get drawn out of a hat to decide who plays who, and the winner goes through to the next round, until you get to the final. Again, each country will have its own examples - the English version is the FA Cup, Spain has the Copa del Rey, Germany has the DFB-Pokal, etc. Some countries will also have 'league cups' which are restricted to entrants from a certain set of levels of the pyramid - in England, for example, the Capital One Cup (as the league cup is currently called) only allows teams to enter from the top 4 levels, whereas the FA Cup invites entrants from all 20+. Most of the national cups will have some kind of weighting, so that top level teams don't have to join the competition until a later round rather than having to spend many extra matches slugging it out with tiny teams.
European competitions - there are only two of these at the moment, the Champions League and the Europa League (which used to be called the UEFA Cup). Entry to these is decided by how well a team does at domestic level the previous season - normally the top 3-6 teams will get to play in one of the continental competitions (the exact number of places depends on the league's size and success, which is judged by a complicated set of coefficients that don't really matter for new spectators. For example, because England's Premier League is big and successful, they get a lot of spaces - 4 in the Champions League and 3 in the Europa League). The European tournaments are pretty big and prestigious, especially the Champions League which is basically considered the hardest trophy for European clubs to win - so it's not just 'taking time off to play some foreigners'! Scheduling-wise, the games are normally in mid-week as well, so they don't interfere too much with a team's fixture list, although they can add fatigue for the players - it's not uncommon to see a team play somewhere in the region of 50-60 games a season if they do well in both a European competition and the domestic cup(s).
The Champions League is the bigger of the two competitions - as the name suggests it's the competition all the national champions will play in, plus a load of second-, third- and fourth-placed teams from bigger leagues. The format begins with qualifying rounds in the summer for the teams from smaller nations, and lower-ranked teams from slightly larger nations, and then the competition 'proper' starts in the autumn with a group stage - this is a set of mini-leagues with eight groups of four teams, placed by a seeded draw - each team will again play the other 3 in its group home and away, with the best 2 teams over those 6 games apiece going through. After Christmas the competition turns into a knockout, although unlike (most of) the national cups every round except the final is done over two legs (home and away), with the winner on aggregate going through.
The Europa League's format is basically the same, with a group stage and then a knockout section; the only real difference is the entrants, who are the teams who finish just below the qualification places for the Champions League as well as national cup winners (so in England, the 5th-placed Premier League team, the FA Cup winners and the League Cup winners get entered into the Europa League).
Hope that all helps!