---Note: Links may contain spoilers!---
Most of the details below are available on UESPWiki under the entry Skyrim:Factions or in pages linked therein.
The only mutually-exclusive factions in the base game are those that represent the two sides in Skyrim's civil war.
It's also worth making mention here that your choice of who to follow early in the game (Ralof or Hadvar), when escaping Helgen, does not determine which of the above groups you are joining.
The two paths essentially merge in Riverwood, when family members of either side will suggest that you take news of Helgen to Jarl Balgruuf the Greater in Whiterun.
The NPC you follow out of Helgen, or one of their family members, will invite you to join their cause. This will add a quest to your journal, but still does not constitute a binding of your character to that faction. Afterwards, you're still free to join whichever side (or none) in the war you please.
A quest to join the opposing side can usually be picked up by simply encountering one of their soldiers in a city or camp that they control. The soldier will suggest that you join their cause, and this will populate that quest in your journal. Once you've expressly declared your allegiance though, you'll fail the quest for the side that you chose to oppose.
You still have one more chance to switch sides though, during The Jagged Crown. Simply deliver the crown to the opposition's leader, and you will join their cause instead.
All that having been said, there's several other factions you can join freely. Joining any of these will not preclude you from joining the others. They also do not affect, nor will they be affected by, your choice regarding the war.
Near or at the end of the quest lines for most groups, you will generally end up becoming the group's leader. This also will not affect your ability to join or lead other factions.
*The quest to join The Dark Brotherhood actually starts in Riften.
**There is (presuming you pursue the main story's questline to its end) an unavoidable quest which puts you in the middle of a conflict between the Blades and Greybeards. Your decision (or even a lack thereof) in this matter will restrict you from certain resources available via the faction you have chosen against. I do not list these as "mutually exclusive" factions though, since you never properly "join" the Greybeards.
The Dawnguard DLC adds in two new player-joinable factions, which are also mutually exclusive.
Note: While these two factions do ultimately become mutually exclusive, you must follow through some of the initial Dawnguard questline before you are given the option to join the Volkihar.