From Halopedia, the Halo wiki
The Connection Host, often shortened to Host, is the console that acts as a server during a Halo 3 or Halo: Reach online Multiplayer match. The other consoles in the game are clients, and rely on the host for synchronization.
Neither Halo 2 nor Halo 3 use dedicated servers—that is, multiplayer matches in those games are not controlled from a server owned by Bungie or Microsoft. Dedicated servers for Halo 3 alone would be far too expensive; instead, a distributed networking model is used. In a distributed networking model, one Xbox keeps track of the game, and the other consoles rely on it to manage things like damage and spawns.
A host is needed because during a multiplayer match, things like lag and latency can cause the consoles involved in the game to "disagree". When such a disagreement occurs, the host's version of events is accepted by all consoles.
Usually, the player with the best connection ends up being the host. However, if a player with a superior connection joins the game after a host has been selected, then they will not become host. If, however, the current host lags out, then the newcomer will become host (assuming that their connection is still superior).
Let's hypothesize that there are three players involved in an online multiplayer match: Person A, Person B, and Person C. Each player has their own console, and Person A is host.
Person A is running into a building with a window. Person B is firing a Rocket Launcher at them, and Person C is standing next to Person B while watching Person A. However, a minor network quirk occurs, and the three players' consoles end up disagreeing on what happened.
A problem has been encountered: each of the consoles have a different version of events. The match cannot continue until this issue has been sorted out. How do the consoles reconcile their differing perceptions?
Person A's console is the connection host, so the consoles being used by Person B and Person C will automatically accept Person A's version of events as being true. On Person B's screen, A's corpse will be replaced by an unshielded but living body; on Person C's screen, A's body will suddenly spark, as if its shields were drained.
Finding the Connection Host
Several glitches depend on synchronization (or desynchronization) with the host; such glitches may only work on the host console. The same applies to some mods. For modders and glitchers, methods of identifying the host can be quite useful.
There are several ways to identify the connection host.
This method is the easiest, but it will only work in Forge. Simply have everyone in the match enter Edit Mode and look straight down. Have everyone return to Player Mode; the player that is still looking down is the connection host.
This method can be used in Matchmaking. Grab any equipment item and walk up to an Overshield or Active Camo without actually touching it. Crouch, and walk to the powerup very slowly while holding RB. If you use the powerup, you are the connection host; otherwise, you will pick the powerup up as if it were a typical equipment item.
Simply bring a party to the Theater lobby. To view films, the host must be the party leader.
Have the Battle Rifle out, and press R+R+Y. Two bursts (six bullets) will be heard coming out of the gun; then, you will switch to the secondary weapon. After that happens, switch back to the Battle Rifle. If it shows that one burst (three bullets) have been fired, then you are not the connection host. If it initially shows that two bursts were fired, but then quickly changes to show that only one burst has been fired, then you are not host. If it shows that two bursts were fired, and continues to show that two bursts were fired, then you are host.
Forcing the Connection Host
It's possible to force a person in the party to be the connection host.
Have everyone except the person who wants to be host leave the game and join back. When the current host leaves, a new host will be selected. Eventually, the person who wants to be host will be the only player left in the game, and they will become host (as they have the best, and only, connection). The game does not reassign host status when the others join back, even if they have better connections.
If you are hosting a server on Halo: Combat Evolved Trial, you may notice that you are a little bit stronger than the other players. You may have also notice that the host is more resilient, especially if you are fighting 1 on 1. This is called Host Advantage, Bungie made this advantage in for anyone hosting a Multiplayer Server. It gives the host a 15% upgrade/advantage. This affects the host's shield strength, health strength, and weapon strength. This doesn't affect their speed, jump, vehicle speed/power, or the amount of power they get from the Overshield/Active Camo.