What part do I need to search for to fix this leaky outdoor hose faucet?
The anti-siphon part on top is broken.  You need to determine the brand and purchase the replacement part for the anti-siphon.
The brand can be seen here.  Take a photo of the brand here.


For the similiar style shown in my comment earlier, a replacement part is sold.  If your faucet is a PRIER or Mansfield, this is the part needed.




You would order a similar part/kit for your brand.

You can unscrew the plastic part,  It is most likely cracked to gush that much water.  Try this universal replacement part shown in this Youtube video. 

Replacing part with universal repair kit

I found this part, which looks identical to your photo. Remove the plastic piece and see if it looks like this.


IF the right blue handle is turned inline with the outlet, water should come out, even if the top cap is leaking.

This part can be purchase here.



You stated you didn't want to replace the whole fixture, It might be easier (@ more cost) to replace the whole faucet with a similar type.
Another solution is to replace the broken faucet with a standard hose faucet, and add a vacuum breaker part on the hose bib end.