The ending shows the Fellowship breaking apart, and the film leaves the Ring quest and the war against Sauron and Saruman open for the next instalment. Boromir tries to take the Ring and then dies defending Merry and Pippin from the Uruk-hai, so the group formed at Rivendell no longer exists by the final scenes.
Why does the film end this way? In my opinion, the ending works because it splits the story into three tracks without resolving any of them. Frodo and Sam cross the river and head toward Mordor by themselves. Aragorn, Legolas and Gimli chase the Uruk-hai to save Merry and Pippin. Merry and Pippin are carried off as prisoners. It could be argued that Frodo choosing to leave alone is a reaction to Boromir's attempt on the Ring, which would make the temptation, rather than the Black Riders, the real threat of the film. Sam insisting on coming along then changes the tone from a solitary burden to a friendship.
The film plants this idea early and pays it off in steps. At Weathertop, Frodo is stabbed by a cursed blade, which shows the Ring has enemies who can reach him. At the council in Rivendell, the Ring is judged too dangerous to keep or use, which shows that even good people cannot be trusted with it. In Lothlorien, Galadriel warns Frodo that the Ring will test the company, and Boromir is the first to fail that test. Each scene raises the stakes a little before the breakup, so the final split reads as the result of pressure rather than a convenient cliffhanger.
The loss of Gandalf in Moria matters here too. Once he falls fighting the Balrog, the Fellowship has no wizard to settle arguments, and Aragorn is left to lead a group that is already fraying. Most of the Fellowship's decisions after Moria are made by people who are tired and grieving, which makes the collapse at the river feel earned.
How does it compare with the rest of the series? Each of the other Lord of the Rings films opens with a flashback or a reminder of the Ring's history, and Fellowship's prologue about Sauron's master Ring is the template. the two towers then picks up with the broken company, with Frodo and Sam continuing toward Mordor and Aragorn, Legolas and Gimli chasing the Orcs who took Merry and Pippin, and return of the king opens on the origin of Gollum, with Déagol finding the Ring in a river. By contrast, the hobbit an unexpected journey begins with the elderly Bilbo writing his story for Frodo, a frame that makes it a lighter, more self contained adventure.
The money is a useful marker of how big a bet this first film was. Fellowship cost $93 million and earned about $898 million worldwide, according to Box Office Mojo, which is a return of nearly ten times its budget.
The wider structure is laid out in lord of the rings timeline, and our lord of the rings movies ranked page gives a verdict on where this film sits in the series.