Henry Cavendish discovered Ohm's law first, but he did not publish his discovery. After several years of experimenting, Ohm published it under his name. Ohm's law appeared in the famous book Die galvanische Kette, mathematisch bearbeitet (The Galvanic Circuit Investigated Mathematically) (1827) in which he gave his complete theory of electricity. He spent several years doing this experiment. At that time, people had no idea about resistance. He began his research with Volta's electrochemical cell; however it didn't work. He improved the experiment equipment by using thermo cell. He also invented the equipment to measure the current. Ohm repeated his experiment by changing the conductor's length. In April, 1826 Ohm published his result:
where 
is the current. 
is the electrical conductivity,
is the section of the conductor, 
is electric potential difference,and 
is the length of the conductor. During the experiment he determined that there is a direct proportionality between the potential difference (voltage) applied across a conductor and the resultant electric current, which we now know as Ohm's law. Now a days, people widely use this law, but at first people didn't believe it.
Georg Ohm
This picture comes from: http://en.wikipedia.org/wiki/Image:Ohm3.gif
Reference
1.http://en.wikipedia.org/wiki/Georg_Ohm