Zaeed is a companion which doesn’t come with the main game. If you want to use him, you have to purchase and download a separate DLC. Zaeed is an ex Blue Suns mercenary who is hired by Cerberus to help you in your quest to save the galaxy from the Reapers.
Like Kasumi, his loyalty mission, The Price of Revenge, is available the moment you recruit him, however I wouldn’t advise doing it straight away, because you’ll meet him right at the start of the game on Omega, when you didn’t even get Mordin. If you don’t have Mordin, you don’t have a tech lab, and you don’t have any upgrades, so it’s best to wait a little later in the story before you actually do this mission.
Notes about the video: I am using Geth Plasma Shotgun. I mainly use Zaeed’s disruptor ammo, it’s the type of ammo that works best with the GPS. Due to the fact that it has only 3 pellets, using any other ammo that has an effect on anything “beneath” the shields won’t have much effect as the shields effectively stop the effect. If you were to fight biotics barriers, you’d use Warp Ammo.
You will see that the GPS will force me to get cover in funny places so that I can charge it. It’s the downside of this shotgun with a Vanguard. But it worked out, so what I’d say is that if your style works with it, then use it. Much in this game comes down to personal playstyle.
The most notable skill Zaeed brings to the table is Disruptor Ammo, which deals extra damage to shields and synthetics. I tend to spec him for squad ammo, since my vanguard only has Incendiary Ammo, so when I know that I will need to kill a lot of people with shields, I will bring Zaeed, activate his squad ammo so that I can use it, and I am all set. Other shotguns will work too but will change your playstyle (for example if you want to use Cryo Ammo you’d use the Scimitar.)
This is what we have from the Illusive Man’s Dossier:
We’ve reached an agreement with veteran mercenary Zaeed Massani. You may know the name; Zaeed has been involved in some of the best known (and some utterly unknown) military operations in the Terminus Systems, and is feared as a ruthless and relentless bounty hunter. I felt you might need a man with his skills on your mission, so I arranged to have him join you.
You will find him on Omega, where he’s wrapping up his current bounty. Don’t worry about his fee; I’ve taken care of that personally.
Suggested companions: Zaeed (mandatory) and anyone you like. Miranda is always the best choice due to her party damage and her ability to deal with anything. Companions with Overload are preferred since a lot of enemies will have shields, but if you waited long enough for Zaeed to have Squad Disruptor Ammo before you did this mission, you could even bring a full biotic such as Jack or Samara.
Mode: Insanity. Class: Vanguard.
Additional info can be found in the Mass Effect 2 Wiki.