Which technology in the past fifty years has changed your life the most?

I think that there are a lot of technologies which have appeared in the past fifty years that make our lives better. Due to my young age it’s hard for me to compare the modern world with the world fifty years ago, but I can show how recent technologies changed my life in the last ten years.
In the beginning, I believe that computers and especially internet had the biggest influence on my life. I remember how in my middle school I had to visit the library to do my homework, because it was very hard to find information that I needed. Sometimes I spent more than three hours per day in library to write a small essay about some historical event. Nowadays the situation is drastically different. All that I need to prepare for exams or make a presentation about some topic is computer and internet access. With the internet and such sites as Wikipedia, I am able to find anything I want from my home or work. Computers and the internet make our lives a lot easier and allow us to have more free time.
The second technology which I should emphasize is mobile phones. I remember the  days when I was young and did not have a mobile. My school was very far from home, so I had to live in dorms. Therefore, to call my parents, I needed to go to other buildings, wait in the queue and only after that was I able to speak with my mom and dad. It was very inconvenient and gave a lot of trouble. Nowadays I can take a phone out of my pocket and here we are. So today it is much easier to communicate with people by phone than it was ten years ago. That is why I think that mobile phones significantly changed my life and lives of other people.
In sum, I would like to say that there a lot of technologies which influenced my life last ten years, but the most changes were caused by computers and mobile phones. Who would like to live without these devices in modern world?

[Excellent for the most part.  Like many Russians, you have a problem with the English articles (a,an,the) and when to use them.  However, your writing is still quite good.]

Which technology in the past fifty years has changed your life the most?

I think that there are a lot of many technologies which that have (there is preference nowadays for restrictive 'that' over 'which') appeared in the past fifty years, that (nested 'that's see my note below) make our lives better. Due to my young age it’s hard for me to compare the modern world with the world of fifty years ago, but I can show how recent technologies have changed my life in the last ten years.
In the beginning, I believe that computers and especially internet have had the biggest influence on my life. I remember how in my middle school I had to visit the library to do my homework, because it was very hard to find information that I needed. Sometimes I spent more than three hours in a per day in the library just to write small essay about some historical event. Nowadays the situation is drastically different. All what that I need to prepare tofor exams or make a presentation about on some topic is a computer and internet access. With internet and such sites as Wikipedia I am able to find anything I want from my home or work. Computers and internet make our lives a lot easier and allow us to have more free time.
The second life-changing technology, I feel which I should emphasize, is mobile phones. I remember days when I was young and did not have a mobile. My school were very far from home so I had to live in dorms. Therefore to call my parents I needed to go to other buildings, wait in the queue and only after that was I were able to speak with my mom and dad. It was very inconvenient and gave me a lot of trouble. Nowadays I can take a phone out of my pocket and there we are. So today it is much easier to communicate with people by phone than it was ten years ago. That is why I think that mobile phones significantly changed my life and life of other people.

In summary, I would like to say that there are a lot of technologies which that have influenced my life in the last ten years, but the most greatest changes were caused brought about by computers and mobile phones. Who would like to live without these devices in modern world?

Your English is really good, the vocabulary is fine, there are just issues with articles, and prepositions - but everyone has such issues.

It is interesting that you have two sentences that contain two 'that's!  The very first sentence is difficult, and feels awkward to me, so I have tried to find a smoother way to express it.  The structure is 'I think that there are [X] that/which [Y] that [Z]'

To simplify, first drop the 'I think that', and break the sentence into two.  How about:

Over the last 50 years many new techologies have appeared.  Of these, some have had significant impact on our lives.