Here I will describe the actual process.

First, you need to download the source code. Manually typing the path on the server is a special kind of entertainment. And downloading from SourceForge via Lynx didn't work.

wget http://sourceforge.net/projects/zabbix/files/ZABBIX%20Latest%20Stable/1.8.6/zabbix-1.8.6.tar.gz

If something is missing, install it. I don't think I forgot anything…

apt-get install mysql-server apache2 php5 php5-mysql php5-gd libmysqld-dev libcurl4-gnutls-dev libsnmp-dev fping snmpd

Add the database, user, and set a password for the user.

mysql -u root -p
mysql> create database zabbix character set utf8;
mysql> grant all privileges on zabbix.* to zabbix@localhost identified by ‘zabbix’;
mysql> exit

Extract the archive.

tar -xzf zabbix-1.8.6.tar.gz

Import the schema and data into the newly created database.

mysql -D zabbix -u zabbix -p < zabbix-1.8.6/create/schema/mysql.sql
mysql -D zabbix -u zabbix -p < zabbix-1.8.6/create/data/data.sql
mysql -D zabbix -u zabbix -p < zabbix-1.8.6/create/data/images_mysql.sql

Then build and install the server.

cd zabbix-1.8.6
./configure --enable-server --enable-agent --with-mysql --with-net-snmp --with-libcurl
make install

Check the services.

cat /etc/services | grep zabbix

The response should be as follows:

zabbix-agent	10050/tcp	# Zabbix Agent
zabbix-agent	10050/udp	# Zabbix Agent
zabbix-trapper	10051/tcp	# Zabbix Trapper
zabbix-trapper	10051/udp	# Zabbix Trapper

Otherwise, add them manually.
Next, create a folder for the configuration files, set the appropriate permissions, and copy the configs there.

mkdir /etc/zabbix
chown -R zabbix.zabbix /etc/zabbix/
cp misc/conf/zabbix_agentd.conf /etc/zabbix/
cp misc/conf/zabbix_server.conf /etc/zabbix/

Edit /etc/zabbix/zabbix_server.conf

DBPassword=zabbix
DBuser=zabbix

I also add the path to the MySQL socket.
Then copy and load the startup scripts.

cp misc/init.d/debian/zabbix-* /etc/init.d/
update-rc.d zabbix-server defaults
update-rc.d zabbix-agent defaults

And now start the server.

/etc/init.d/zabbix-server start

All that's left is to set up the frontend.
Create a directory, copy the scripts there, and set the owner.

mkdir /var/www/zabbix
cp -R frontends/php/* /var/www/zabbix/
chown -R www-data:www-data /var/www/zabbix

Finally, edit /etc/php5/apache2/php.ini and restart the web server.

post_max_size = 32M
max_execution_time = 600
max_input_time = 600
memory_limit = 256M
upload_max_filesize = 16M
date.timezone = «Europe/Moscow»