Terminals 1-3 are situated on level 6, ‘The Ark‘

Terminal 1 The first time you go inside, you will see a door almost immediately on the left. Simply enter this area of the level to find the first of the seven terminals.

Terminal 2 When you go inside the structure to let the tanks across the bridge of light, there will be a Terminal almost directly behind the controls for the bridge.

Terminal 3 You’ll miss this terminal if you are not careful. After defeating the scarab, you will head into a large structure. There will be a room with several sleeping grunts. Nearly directly following this room is another room in which you can take two different directions. There is a door, and a staircase (both eventually lead to the next room). Underneath the staircase is an additional door that leads to the 3rd terminal.

Terminals 4, 5, and 6 can be found on the 7th level, ‘The Covenant’

Terminal 4 When you access the first Elevator leading to the tower controls, jump from the elevator platform to the platform on the opposite side from where you accessed the elevator from. The terminal is dead ahead.

Terminal 5 After getting a Hornet vehicle, fly towards tower 2 (the one that the Arbiter has already disabled). A voice will remind you that Tower 2 is already clear, just to confirm you are at the right place. The Terminal is just inside of the structure.

Terminal 6 Terminal 6 is identical to Terminal 4, only in the 3rd tower. Just jump to the platform, and there it is.

Terminal 7 can be found on the final level of the game, ‘Halo‘

Terminal 7 Terminal 7 is found at the beginning of the final level in Halo 3. To find it, simple head forward from the start of the level, until you encounter the area with a few structural beams going from one wall into another. At this point, turn right, and look for a beam separate from the rest. Follow this beam into a doorway. Cortana will ask you, “Where are you going?”. Turn left once inside, and the Terminal should be in full view.

Source: Wikia