Installed on the second host prubuje MySQL Workbench and connect to the database. Please help me.

..and the problem is?

The problem is that, while the workbench after installing the program on a second computer can not connect to the database that is installed on another computer. Starting "new server instance" I select "remote host" I enter the IP address push "Next" fill "UserName" and push "Next" when testing the database connection is created in the Message Log entry "Can` t connect to MySQL server on 10.120.120.184 ".
I made available in the network a folder where I installed MySQL database. I do not know if this is necessary, but that did not help.

My suspicion is the firewall on the server is blocking the connection so make certain the firewall on the computer with MySQL on has port 3306 open

first off from the client computer ping the server address
ping 10.120.120.184
..if the response is 'reply from......'
then we know that the client computer can 'see' the server computer
if the response is 'destination host unreachable' then you have a network communication error. ie the two computers cannto see each other and cannot communicate. if so you need to work out why

if you have telnet on your computer use it with the address 10.120.120.184:3306
if you don't then either download telnet and run it
OR
post 10.120.120.184:3306 in your browser's address bar, that should download a file, examine trhe contents of that file

I don't think its a user permissions problem yet. but if it is I'd strongly recommend that you read up on how to set up user accounts, hosts and permissions

Does it matter how to install MySQL? During installation, there were two choices. Do you know what the installation possibilities are available? Perhaps I should uninstall and re-install the database in a different mode?

"Can not connect to host on port 23". Why is there port 23?

Try

telnet host_or_ip_address 3306

I'd check the firewall on both computers to make certain telnet can communicate of port 23

how did you install MySQL?
most of these problems can be short circuited installing using XAMPP. given that you seem to be struggling with some of the basics I'd suggest you consider removing the current MySQL installation and install XAMPP.
take a backup of your current table design.
download XAMPP for your operating system (where the server is to be loaded)
install XAMPP
take note of any security or other post installation issues
then repeat the process
validate the server is running on the computer hosting the server
validate MySQL is accessable from the client computer

You are using telnet incorrectly. You should not include a : with the port number. The hostname or ip address and the port number are 2 separate parameters. Port 23 is the default port for telnet.

XAMPP 1.7.2 includes MySQL version 5.0.33. And I have installed version 5.5.2 and so I wanted to have a version of it. Can I install XAMPP I can only have the older version?

System turned off the firewall on both computers. I typed the command without the ":". This result is the same: "connection to host lost". But I suspect the mysql service.

looking at the XAMPP site MySQL is on version 5.5.1.6 for all installations except Mac which are 5.1.44. there are some significant bug fixes between 5.0/5.1 and 5.5. so I don't know where you are getting the 5.0.33 version from

unless this is a production server or your use is data warehousing then any of those versions will suffice.

right now your problem is getting your client computer to talk to the server computer. its clear you are struggling with gettignt hat to work, whether thats a language issue (ie Ronan and myself are incomporehensible) or a knowledge / skills issue I don't know.

XAMPP will install a fully working MySQL server
it will install other tools
it will make your life easier to get the system working

once its working then you can update the MySQL server if you so wish or need. however right now the aim is to get your server working.

it may be a smarter use of your time to find someone near you you who can help you install the software and make certain the netwrok works.

we don't know what operating system you are installing on
we don't know what you are doing or trying to do
frankly its tough to diagnose these sort of problems remotely