Your arrest is part of your criminal history sometimes referred to as your rap sheet. If you were arrested but no charges were filed you may be able to get the record sealed under PC 851.8. This procedure is difficult and may disadvantage you further if you lose. You can more easily obtain a court order that this was not an arrest but a detention.
You have an arrest record that can be sealed if a judge would find you "factually innocent" - that's hard to get. What was your blood test or breath test result - did DMV take your license?
The arrest will likely show up on background checks. However, California does allow for an expungement of your record. Of course more details are necessary and there are no guarantees.
I recommend hiring an attorney to start the process of expunging your record sooner rather than later, as you may be looking for work now or in the near future.
I am available to help you with this. Feel free to contact my law office at 424 274 1846.