Ahem, I have worked for a masonry company for about 5 years during the summer (owned by my father who has been doing it nearly 40 years).. thus I have some experience in saying that tuckpointing will not fix that problem (atleast not from what I see here) I believe you should hire/find an experienced contractor in your area, they will give you a free estimate.
IMO that entire section needs to be taken down and re-laid at best, at worst there are some major structural issues.
If you are looking to just doctor it up, you can go to Lowes or Home Depot and purchase a bag of quikrete mortar (type S should work
http://www.lowes.com/pd_10391-286-11...ar$identifier=) mix it according to their directions. You will probably need to purchase a couple pointing tools to accomplish this, you will two tools something slightly smaller in diameter than the bed joint (horizontal) and the head joint (vertical). Basically you just hold something flat up to the brick, put mortar on it and push it in the crack until it is even with the surrounding area. You can let it sit for ~20 minutes, at that point you will rake it by dragging the handle of the pointing tool over the joint to knock off the excess. Let it dry a bit, maybe a few hours and hit the brick with a wet sponge to get rid of the discolored spots.
I recommend that before this, you take the time to scrape/brush out the empty joints to assure proper bonding.
To fix any mortar that has the wrong color: it must be cut out with a 6" grinder with a diamond blade to re-point. To match the existing color you can purchase a mortar color.
Honestly though, most masonry tasks like this are way out of the do-it-yourself league and require a skilled tradesman.
Take any advice here at your own risk.