I just can't insert a tampon. How do I do it?

This is a bit awkward so I suggest that anyone who isn't a woman please stop reading now. Way too much TMI in this answer.

It takes practice and you have to be comfortable trying to do it. It's ok if you aren't successful the first couple tries. It's a strange process.

First, have any pants, skirts, and underwear completely off. Once you get used to it, it's a piece of cake to do half dressed but when you are starting, just strip down to basics. Then squat slightly. Unwrap the tampon and hold it in the middle (there are usually ridges at this point. It's where the 2 pieces of the case come together) between your thumb and middle finger with your index finger on the end with the string. Use your other hand to spread your labia (the lips) wide. Take the tampon and poke in the correct area until you find your vagina opening. Now, you want to hold it straight up and down. DON'T. That is the totally wrong angle and it won't work. You'll hit your pubic bone and it won't go in right. Angle the hand holding the tampon forward slightly. Now slowly try to insert the end of the cardboard/plastic case. Do not push the plunger yet. Once you get the angle right, the casing will slide in. When you are in up to your fingers holding the case, make sure your angle is still good and push the plunger to put the tampon all the way in. You will feel it insert. You should feel it way up there. Sorry, that sounds gross and weird but it's the best way to describe it. Take out the case and squeeze your vaginal muscles slightly. You shouldn't feel any obstacles.

Congratulations! You have learned to use a tampon correctly! The more you practice, the easier it will become.

Sonnet, Clare, Melissa, Heather, and many others have given excellent answers about technique and alternatives, and I've been waiting for someone to mention that, if it's not your technique, there are two other possibilities.  One is that your hymen might be so broad that you can't get a tampon past it.  The other is a really annoying problem called vaginismus

Let's start with the hymen.  This is a stretchy bit of tissue that most girls have that partly blocks the vaginal entrance.  Usually it's a fairly narrow crescent like the first drawing below. 

Some girls are born with no hymen at all.  Quite a few girls have hymens that are quite thin and easily become torn, leaving just little "tags," like the ones in the 2nd drawing.  Even an "annular" or ring-shaped hymen, like the one in the third drawing can sometimes be stretchy enough so it isn't a problem. 

But about 1-2% of girls have something like one of the last six sketches, and that means they can't use tampons without first tearing the hymen or getting a doctor to remove it.

Honey, sit on the toilet and open your legs wide. It's scary at first. I was in France the first time I did it and  I was 14. I was lucky and had three of my close girlfriends outside the stall; they said: put your fingers around yourself and open it up; wiggle wiggle wiggle  it back and forth and try to relax your muscles.  Push it SO gently inside. All of the way until it feels a bit funny. Then stop and push the tube gently inside.  If that doesn't work, take it out and put some vaseline on it and try again. Kind of icky. Here we go.

Edit: Many have pointed out that vaseline is not the best thing to use! Am dating myself! Go with the KY jelly!
Don't be embarrassed -- because we all have trouble with this.

All of the other advice here is fantastic.  But I want to add my story and tell that it took me a week to get this, and my main problem was lubrication.

First - absolutely make sure you are using a plastic applicator (not cardboard) and look for "slim" or "light" tampons for your first try.  You can always adjust up later, once you get a feel for it.  Don't try it right after a shower/swim as this will strip away your natural moisture and will make it more difficult for you.

If you are having trouble relaxing, and if you really feel like you are having to force the tampon, try adding a little lubrication. Someone suggested vaseline, which can be ok if you use a tiny bit, but since it is petroleum-based it really isn't the best thing for you.

If you can get it -- something like KY Jelly or astroglide, which is water based and specifically made for use in your vagina, is better for you.  I was too embarrassed to try this when I was 13, so I used a tiny bit of olive oil, which was a little messy, but worked well enough for me to figure things out.

Second - if you have tried everything and it's still not working -- it's not uncommon for women to have different (quite normal) anatomical variations like a tilted or retroverted uterus that can make inserting a tampon difficult or very uncomfortable.  You should talk to your doctor about this next time you go, and they may be able to provide additional advice -- especially if your difficulty is anatomical and not with your technique.
To the good advice already listed, I'd add:

  • If you're having trouble with finding the right spot for inserting, read the instructions that came with your tampons carefully, and look closely at the diagram.   Grab a hand mirror and make sure you locate the vaginal opening.  Check with your (clean!) finger - sometimes you need to open things up a bit.

  • It's easiest to do this on a heavier flow day.

  • Use the smallest size first, but keep a pad on as well until you get the hang of it.  You don't want to miscalculate and have to change your clothes.

If you're really striking out, it might be worth a trip to the gynecologist.  You should be getting checked annually anyway.
I've never had to explain the process, but havin. 12 year old daughter at home. I may have to soon. Your best start is a slim/junior size, with a plastic applicator (at least for your first few attempts). The next is to relax!!  Really really relax, you aren't going to do any harm if you don't succeed at first. Privacy would help. The other good suggestion is that if you're right handed, put you left foot on the toilet seat. With the tampon in your hand find the entrance with tip of the applicator. You will now only need to put it in slightly, and the plunge the applicator in. You should be left with an empty applicator in your hand.

It's hard thing to give direction for apparently. Best of luck to you!
James LaJocies has a good point, but to that point I'd like to add that if you feel you need a lubricant to insert a tampon, you should really be wearing a pad.
Being too dry to insert a tampon usually indicates that you are not flowing heavily enough to use one and you should just stick with a pad.
I haven't seen this direction yet, so I'll add it to the other, more thorough ones here.

Aim and angle are oddly crucial for getting a tampon placed right. You wouldn't think they would be, I mean, where else can it go in there? But if your angle is wrong, it won't go in right and will hurt.

Aim for the curve of your lower back. Not up towards your stomach, not straight back towards your bum, but at a 45 degree angle towards your lower back. That should get it in the right way.
Could Brazil invade and conquer Venezuela if Jair Bolsonaro was elected and teamed up with Trump?

The Brazilian communication media has been opposing Jair Bolsonaro all the time and selling a fake image of him.Bolsonaro has been a military man in his youth and after that he has been a congress man last 27 years. By the way,

As Apple continues to merge features of iOS and macOS, does it make sense to merge their respective App Stores?

If it's done well, yes, it does, or at least, have more compatibility between them.One of the major problems is In-App purchasing, if there is an App for Mac and iOS, and you buy an In-App purchase on the Mac one, you don't get it on the iOS one, the App Stores are separate, and

Could humans ever use technology in the future to become telepathic?

Sure, not even in the so distant future. By reading and translating the minute muscle contractions by speech muscles like the larynx you could technically digitally send voluntary thoughts to another person.When you have inner speech/subvocalization the muscles move even if you don't actually articulate. With enough advancements you