My chain is catching on a part for which I don't even know the name

Question

A couple of days ago I was riding my bike and when I hit a hill and was shifting down through my gears, the chain slipped off the gears. I hopped off the bike and walked it back home, flipped it upside down, and got the chain back on the gears. I figured maybe I was shifting too quickly through the gears or something.

Yesterday I went out to ride again, and shortly after I started - this time on fairly level ground - the chain slipped off again. I put it back on the gears again and rode a bit farther, but yet again it slipped off.

This time when I took it back home (and yet again got the chain back onto the gears), I tried to carefully troubleshoot what was going wrong. I noticed that my troubles always started when I tried to take the left shifter to 3rd (it's an 18-speed mountain bike) or 1st. When shifting to 3rd, it wouldn't transfer to the gear (stayed on 2nd) and when going to 1st it would come past 1st next to the pedal hub. It might help for me to mention that the right shifter was going through all 6 back gears just fine through all this - so I was getting the idea that my problem had something to do with the left shifter only. After manually applying the chain to the 3rd gear, I slowly turned the pedals and noticed that the chain was rubbing up against and sometimes getting stuck on the part in my photo that I've highlighted with a red circle and for which I unfortunately do not know the name (since I'm a bike noob).

My best guess as to the function of this part is that it supposedly helps guide the chain into the front gear set (at the pedals) - but in my case it seems to be interfering with the chain's movement. My questions are as follows:

1. What is this part's name and function?
2. Should I remove it or adjust it?
3. How can I best go about this?

If I had the money I'd just take it to a bike shop and watch them fix it, but I'm broke so unless they'd do it for free that's unfortunately not an option. As I've already indicated, I'm a total novice at bike ownership so I'll appreciate helpful, constructive replies. I look forward to getting this resolved so I can get back to riding. Thanks for your time.

--------------------- EDIT ---------------------

Operating perhaps prematurely, I tried adjusting the positioning of the derailer to stop it from inhibiting chain movement. I noticed the whole assembly was attached to the post via the screw highlighted in the following picture:

I carefully loosened this screw and then very slowly rotated the derailer away from the chain until it had about a millimeter or so of clearance. I then tightened the screw and tested functionality. Now the pedals ran smoothly with the front gearset in 3rd. I took the right shifter down a couple of gears to verify its continued functionality (all was well), and then I tried taking the left shifter down to 2nd. To my dismay, the chain transferred to 1st gear instead - so I believe my configuration is still amiss.

Here are two better shots (with the bike in upright orientation, post-adjustment) of the derailer:

Answer 1

1. The part is a front derailleur. It job is to shift the chain between the front gears. Change gears while looking at it, you'll notice it moves and pushes the chain between sprockets
2. You need to adjust it, removing it would actually require more effort since it will need you to delink the chain somewhere. Also, removing would remove your gear shifting functionality.
3. What you're describing is a common thing that happens to drive trains over time. The cable attaching your gear shifter on your handle to the derailleur near the pedals, has either stretched if it was low quality, or slipped a little at the derailleur. The fix is quite simple. All you need to do is locate the point at which your cable attaches itself to the derailleur(you can do it by following the black tube coming out of your shifter on the handlebar, along the frame of your cycle and ending somewhere opposite the derailleur on the other side of the cycle) It looks somewhat like this- .
   Shift your bike down to the first gear. Then locate the screw that holds the metallic cable coming out of the aforementioned black tube and loosen it enough that the cable can slide around just a little(In the image above its the screw which the right end of the double arrow is pointing at).
   This is the tricky part, you might want to ask a friend to help if you don't have fancy tools. Move the derailleur like as if it were shifting the chain to the second gear.This might bend your chain a little, that's OK. Notice the stretch of cable visible and marked in the picture would have become noticeably slack. While still holding the derailleur in its forced position, pull your cable taught by pulling the end beyond the screw. Hold the point where the cable is attached with your fingers while letting go of the derailleur. Make sure you don't let that point move, it will take some effort, they make the springs rather tight. Now slowly let the cable loose again by moving the attachment point in the direction the spring is pulling it till the derailleur is exactly centered on the chain, a little closer to second gear position wouldn't hurt though. Holding the attachment point in position, tighten that screw holding the cable.
   As a safety measure so that the next time your cable loosens, the chain doesn't fall off, you could adjust the minimum position screw as described below-
   Flip the bike right side up. On the top of the derailleur attachment, you'll find two small identical screws right next to each other. Something like this-
   
   These screws prevent your derailleur from moving beyond its maximum and minimum positions, it seems in this case one of them came loose and shifted letting the derailleur push the chain off the sprocket.
   With the cycle in third gear, locate which one of the screws is closer to the obstruction below it (You can tell what this obstruction is by forcefully trying to shift past the third gear like as if there were a 4th gear. You'll notice the obstruction in question is pressed against the end of one of the screws when you do this). Once you've done so, pick the OTHER screw and tighten it till the point that it just touches the same obstruction when the cycle is in first gear.

You're all set to go now

EDIT: Since I'm a noob to this forum and don't have commenting privileges yet, I'll put my comment to your update post here.
I only saw the original post which is why i was confused about what was wrong and I mistakenly assumed it to be the extremely common problem of the cable slipping a little due to use. What you did in the update to your post was actually the correct solution, just not enough. Loosen the screw holding the derailleur to the frame and turn it a little more till the derailleur is inline with the chain(not just clearing it). If the inner edge of it is touching now, follow my instructions above for changing the lower limit, but change the upper limit instead (ie the screw which is closer to the obstruction in the top gear) till the derailleur is centered on the chain in third gear.