(covers information from several alternate timelines)
Maxwell Forrest was a Human Starfleet official in charge of the NX Project during the 2140s and 2150s. As a commodore, he supervised the first tests of the warp five engine. As such, he worked closely with the Vulcans, who were supervising Humanity's progress into space.
NX Project Edit
As a Starfleet commodore, Forrest was responsible for overseeing the NX Project and did so from as early as 2143. In that year, he chose A.G. Robinson over Jonathan Archer to pilot the NX-Alpha. This decision was a difficult one for Forrest, since the engine of the NX-Alpha had been designed by Archer's father, Henry.
Robinson easily broke the Warp 2 barrier, but the ship exploded shortly after reaching warp 2.2. Robinson survived by jettisoning an escape pod while at warp. The Vulcans claimed that the warp program should be postponed. Forrest stopped at the 602 Club, a bar frequented by Starfleet personnel, and informed Archer and Robinson that the program would be suspended. After Archer and Robinson disobeyed orders and ran a new test that was successful, Forrest persuaded Starfleet Command to reinstate the program. (ENT: "First Flight")
The Enterprise mission Edit
In 2150, the final selection for the captaincy of Enterprise NX-01 took place between Captains A.G. Robinson and Jonathan Archer. Although Captain Gardner had originally been considered the most suitable choice by Vulcan Ambassador Soval, Forrest – by then an admiral assigned Jonathan Archer as the commanding officer of Enterprise in October that year. (ENT: "First Flight", "Shadows of P'Jem")
After Klingon courier Klaang crashed on Earth during the Broken Bow Incident in April 2151, the Vulcans wanted to delay the launch of Enterprise and return the Klingon to its homeworld: Qo'noS. Forrest overruled them and decided that Starfleet itself would return the Klingon. He authorized Enterprise to launch early, on a mission to return Klaang to Qo'noS alive, despite protests from Soval and the other Vulcans. Forrest told Archer that Humanity had been waiting a hundred years for this chance, and told him not to screw up. Forrest also officiated over the launching ceremony of Enterprise. At a press conference immediately prior to the mission launch, he said that Humans had been ankle deep in space travel and that it was time to swim. (ENT: "Broken Bow")
The mission Edit
Forrest stayed in constant touch with Enterprise. He gave many orders as different situations arose.
He ordered the ship to investigate a distress call from ECS Fortunate, a cargo ship that was attacked by Nausicaan pirates. (ENT: "Fortunate Son") While Enterprise was visited by a group of Vulcans who did not embrace logic, Forrest contacted Archer and asked him to have one of the Vulcans, named Kov, call his father, who was dying. (ENT: "Fusion")
Forrest had to constantly walk a delicate balance between the goal of Human space exploration and Vulcan misgivings about it. After the discovery of P'Jem as a base for spy activity against the Andorians, it was destroyed in an attack by the Andorians. The Vulcan High Command called for an end of Enterprise's mission. Forrest refused, so the Vulcans called off joint Earth-Vulcan fleet operations and requested Sub-Commander T'Pol to transfer off Enterprise. The Vulcans eventually relented after T'Pol and Archer saved Sopek, a senior Vulcan officer. Later, Forrest ordered Archer to arbitrate a dispute between the Andorians and Vulcans after hostilities broke out over the disputed planet called Weytahn. He also ordered Enterprise to transport Vulcan Ambassador V'Lar from the Mazarite homeworld. She had information on the corruption of government officials and her life was in danger. (ENT: "Shadows of P'Jem", "Cease Fire", "Fallen Hero")
After an accident seemingly caused by Enterprise killed 3,600 Paraagans when tetrazine was ignited by the ship, Forrest informed Archer that at the urging of the Vulcan High Command and Starfleet, Enterprise's mission was canceled. After Archer presented evidence that vindicated Enterprise, Forrest reinstated the mission. (ENT: "Shockwave", "Shockwave, Part II")
After Archer contacted him about the discovery of a time travel pod which had a Human corpse in it, Forrest thought it might be Zefram Cochrane, who had disappeared mysteriously. (ENT: "Future Tense") After a science team in the Arctic Circle lost contact with Starfleet, Admiral Forrest was informed and led a team to investigate. They rushed to the site, only to find it abandoned. He found that the scientists were missing, as was their transport. Admiral Forrest called in Enterprise to find the transport. This was Humanity's first contact with the cybernetic beings known as the Borg, although the Humans never found out what their name was. (ENT: "Regeneration")
The Xindi mission Edit
After an initial Xindi attack on Earth and once Archer met with a humanoid from the future who gave him details about the Xindi, Forrest met with Archer at Starfleet Headquarters. At first, Admiral Forrest was not convinced about Archer's information. Forrest was skeptical that it justified a mission into the Delphic Expanse. After Archer provided evidence by proving that the remains of a Xindi probe that had carried out the first attack on Earth included some debris which was from the future, Forrest authorized Enterprise to depart on a mission into the Delphic Expanse and to conduct a search for the Xindi there. (ENT: "The Expanse")
Later career Edit
After the Xindi mission succeeded, Forrest welcomed the Enterprise crew home as heroes. Archer was debriefed by Starfleet Command and the Vulcan High Command. When Ambassador Soval insinuated that trellium-infected Vulcan ship Seleya was destroyed without the Enterprise officers doing enough to help the Vulcan crew, Archer became outraged, blaming the Vulcan High Command for not providing enough assistance in their mission. Admiral Forrest ordered Archer to take some time off to clear his mind. (ENT: "Home")
Forrest visited the United Earth Embassy on Vulcan in 2154. There, he had a conversation with Ambassador Soval, discussing joint missions between Earth and Vulcan. When a terrorist bomb detonated in the embassy, Forrest pushed Soval out of the blast wave, sacrificing himself. This proved to be Forrest's last act, as he was killed by the explosion. Captain Archer was saddened by the loss of his friend. Ambassador Soval, for his part, offered the Captain his full support in the investigation and urged him to look on Vulcan for the answers he needed. (ENT: "The Forge")
In an alternate timeline, Forrest sent a message to Enterprise after meeting with the Command Council to transfer command of the ship to Sub-Commander T'Pol. He was likely killed when the Xindi weapon was deployed against Earth, shortly thereafter. (ENT: "Twilight")
Background information Edit
The character of Forrest was named after DeForest Kelley, the Original Series actor who played Doctor McCoy.  Forrest's first name, "Maxwell", wasn't established until ENT Season 4 installment "The Forge", the only episode that uses that first name.
While Rick Berman and Brannon Braga were developing Star Trek: Enterprise pilot episode "Broken Bow", they had no idea that Forrest would become a long-term recurring character. ("Broken Bow" audio commentary, ENT Season 1 DVD/Blu-ray) Hence, in a character breakdown sheet that Paramount sent to various talent agents upon casting "Broken Bow", Admiral Forrest was described as a non-recurring role, scheduled to appear only in the pilot episode. He was, in the same document, further characterized thus; "Human male. 50s to 60s. A career military man who's the highest ranking officer in Starfleet. He's fond of Captain Archer and has personally selected him to command Enterprise." 
Vaughn Armstrong found adopting the role of Forrest, for ENT pilot episode "Broken Bow", was somewhat odd, as he was much more used to playing aliens. Regarding the character's debut, he remarked, "It was great! Though I didn't have a lot of time to memorize my lines, because I usually do it in the makeup chair [....] This time I could actually sit in my trailer and learn my lines, but it was unusual in that there wasn't anybody patting my face while I was doing it!" (Star Trek: The Magazine Volume 2, Issue 9, p. 42)
Following the initial appearance of Forrest in "Broken Bow", Vaughn Armstrong suspected the character would return. (Star Trek: The Magazine Volume 2, Issue 9, p. 47)
In the first draft script of "Fortunate Son", Forrest had more dialogue than he does in the final version of that episode. For instance, he wrily mentioned that he'd been told a particular comet would be visible from Earth in approximately ten thousand years. He also said the ECS Fortunate was "pretty far out there" and asked Archer to "keep those 'postcards' coming" regarding scans from space which Enterprise had been relaying back to Starfleet Headquarters, on Earth.
Forrest was referred to in a deleted scene from "E²". In that scene, Phlox recalled that Forrest had allowed Captain Archer to bring Porthos aboard Enterprise, Phlox citing this as potential evidence that Starfleet might allow children on-board starships someday. Archer then imagined "the look on the admiral's face when he sees two Enterprises pulling into spacedock." (ENT Season 3 DVD & Blu-ray special features)
In the final draft script of "In a Mirror, Darkly", Forrest was described as "heroic" and "selfless".