@brokeu91
The cup games are games played between clubs. Generally speaking, all the clubs' names are thrown in a hat and drawn out one by one (home team v away team and so on) until an entire round has been decided. Once those games have been played, the process is repeated for the next round of the cup. There is no seeding so the two best teams in England could face each other in the early stages of a cup competition. Teams don't qualify as such for a cup competition within a country in the same way NFL teams will qualify for the post-season. They will be placed in the draw for the first round of a cup simply by existing (although the bigger clubs tend to enter cup competitions later than smaller, poorer teams in lower tiers. The soccer league system resembles a huge pyramid with the professional leagues at the top (the Premier League being the top of those)
However, teams do qualify to take part in the Champions League which is a cup competition where club sides across Europe compete agianst each other. In England, the teams that finish in the top four of the Premier League qualify for the following season's Champions League. The Champions League is a more prestigious competition to win (in mine and I think most people's view) although fans of the top teams would expect their team to challenge for both the Champions League and the Premier League (or the equivalent in whichever country they're from).
It does suck to see your team relegated from the Premier League down into the second tier. I've witnessed my team do just that on three separate occasions in the past few years (we're what's known as a yo yo club). In fact, the last time we were relegated was at the end of the season 2010/2011. During that season we also won one of the major cup competitions at Wembley Stadium (bitter sweet).
International games are, obviously, games between countries. Countries will play a number of friendlies and competitive games throughout the season. The competitive games will be qualifiers for either the European Championships or the World Cup (both are scheduled so that there is a tournament every two years during the summer break between domestic football/soccer seasons. The England manager (for example) will choose a squad of English players that he feels are the best (at that point in time) and the domestic soccer schedule has built in breaks to allow these internation games to take place. The international tournaments begin with a group stage and the top two teams from each group then go into a knockout stage similar to the NFL play offs. A tournament is played over the course of one month or so.
Players are bought and sold between clubs with cold hard cash (sometimes with other players included as part of a deal). Each club is responsible for its own player development and will sign up kids aged as young as 7 to learn under their youth coaching systems. A club's youth teams will play in a league against other clubs' youth teams. There is also no draft or parity like in the NFL. The clubs with the most money will (most of the time) rise to the top.