BONUS! I've also included an extra video (by Nak3d Eli) for the 11th terminal through Halo Waypoint, which nets you 100,000 free credits. It isn't required for the achievement.
You need to be in HD mode to see the terminals. (Press "select" if you're in location and the terminal isn't showing.) You don't need to watch the video at all—just enter it. (You can always view them from the Extras menu.) Press your action button to use the terminal (X in regular mode, B in Bumper Jumper).
Terminal 1: Pillar of Autumn
Upon arriving at the bridge (where you meet Keyes), walk around the left side for the flashing red terminal. If you miss it, you can return after Keyes shuts up, or even after receiving the pistol.
Terminal 2: Halo
When in the underground complex in the Warthog before making the gap jump, a hallway glows with a yellow terminal. It's hard to miss if you just follow the path.
Terminal 3: Truth and Reconciliation
In the bridge room (with the elevated platform with lots of screens) before reaching the jail cells, the terminal is the red screen directly ahead on the platform. For an audio cue, Cortana will say that Keyes's "transponder signal is strong."
Terminal 4: The Silent Cartographer
After passing into the path in-between the rocks en route to the override, the clearing with the two Hunters (big spiky guys with who shoot green Fuel Rod shots) holds the terminal. It's blue and up against the left side of the central structure's pillar.
Terminal 5: Assault on the Control Room
At the end of the level with the pyramid-like structure, make a left towards the crag in the ground instead. The blue terminal lies against the back of the base of the gray structural support entering the ground. (If you steal the Banshee earlier for the achievement, you can fly directly to the ground to skip the elevator segment.)
Terminal 6: 343 Guilty Spark
Read this first. There are three elevators in this level—the one down into the structure, the one that's broken, and the one that goes up at the end. After activating the FINAL lift (up), run back for the entrance whence you came. The yellow terminal is atop that doorway structure.
Terminal 7: The Library
On the third flood of the Library with the gigantic circular void in the ground, you'll see a yellow light flashing across the way. Jump over the spokes around the void to reach the yellow terminal.
Terminal 8: Two Betrayals
At the beginning of the level, walk to the other end of the circular platform for the yellow terminal.
Terminal 9: Keyes
In the bridge room (from terminal three), enter the trench in the ground closest to the jail cells. What do you know? A red terminal has sprouted!
Terminal 10: The Maw (last one)
In the room similar to the one that started the game, after climbing the ladder and encountering the Sentinels (flying laser bots), check out the technology where the Sentinels were flying: the final red terminal.