I was wondering what the best way to deal with enemy champions that are considered "Pub-Stompers". My friend told me that usually these champions are counter-picked or the opposing team can invest in more hard cc. However, I am only level 10 and was wondering if there was any way to beat them without counter-picking.

Examples: Master Yi, Katarina, Tryndamere, Mordekaiser, Akali (don't see often), etc.

Hard CC really isn't counter picking. Your team should have some hard CC as a general rule. Just to shut down channeled abilities or make ganks/teamfights a bit easier if nothing else.

I wasn't completely referring to hard CC being used as counter picking, but in lower level games most of the champions available are free week champions. If you are trying to stop a Tryndamere and a Katarina on the same team this week, I think the only option would be to hope the Amumu saves his stuns just for them.

I mention counter-picking more for champions such as Mordekaiser because of his insane farming abilities or Katarina who destroys skill shot champions. (personal opinion)

Buy wards and communicate. Most of the champions that you listed thrive on the ability to roam and gank. Also, peel for the ADC.

Play with friends... most pub stomp champs are champs that take advantage of poor communication on the enemy team. Warding and communicating among your team will shut these champs down.

This is 100% correct. Wards will save lives. I know, especially early on, that they may seem useless, but try playing a support role, getting like 3 GP10 items, and warding everywhere.

Tips for when you're doin not so great.
1. Tell everyone to play as defensive as possible
2. Ward major locations and make sure you don't get ganked too often
3. Try to communicate your own ganks on weaker members of the other team.
4. If you're really desperate, get Gold per 10 seconds items. Seriously. Stall. This'll work especially if their fed player's inventory is maxed and he can't buy anything more. They'll stay the same strength and you can get stronger.

They are called "pubstompers" because they will really take advantage over teams who don't know how to play against them or are disorganized. I would simply advise you to learn about the champs you mentioned, even play them if you have a chance, because then you will have a better understanding of their mechanics. Particularly when playing as these pubstsompers, take note of their cooldowns and what phases they are the strongest in (early game, mid game, etc). That is how I learned how to deal with just about any Tryndamere, I know that his Ultimate lasts for about 7 seconds and I can simply avoid him during that time, then walk up and ignite/exhaust him to finish him off.