Before Ichigo was born, his mother?s soul was infected with a Hollow from one of Aizen?s experiments. His dad, Isshin, is a Soul Reaper who was the 10th division captain, and he was fighting both Aizen and some oddly powerful Hollow and was on the losing end before his mother saved Isshin. She unknowingly got her soul infected in the process. The merging of a Hollow with another soul is Hollowfication.
The Quincy species isn?t able to withstand any type of Hollowfication, and since his mother was a Quincy, this Hollowfication would have killed her. However, Isshin volunteered to give up his powers to contain the Hollow and link their souls together, and both of them fell in love and later married each other. They lived as a normal, powerless human couple. Ichigo?s mom later gave birth to him.
Now, what happened is that since Ichigo was the firstborn, he inherited a combination of his dad?s Soul Reaper powers and the hollow that these powers contained within his mom?s soul. He also inherited his mom?s former Quincy powers. Hollowfication was a big failure, but Ichigo could be seen as the most successful byproduct of such an experiment because his already-strong Soul Reaper powers were heightened considerably due to these powers fusing with the Hollow that he inherited.