What speaker/output can I run BB through?

Does is run through your guitar amp, computer speakers or mixer/amp and P.A speakers so I can have nice loud drum volume?

BeatBuddy has three sound output sockets - headphones (small 3.5mm jack) and a pair of two large “standard” 6.35mm jacks for two channels of stereo sound. And just as usual, you can use only one of them (namely, left channel) to get mono sound.
Yes, you can then use any mixer, power amp or computer speakers to get BeatBuddy output sound just as well as you can use your headphones for that.
While you can definitely use a dirty channel of your amp together with a BeatBuddy, it should be noted that BeatBuddy sounds best when your power amp is clean.

Beatbuddy will sound best through a full range speaker like a P.A. Guitar amps typically don’t cover the whole spectrum of high and low frequencies. I think a good idea would be to listen to it with a good set of headphones and then try to reproduce that sound with a good set of speakers.

Read the thread Optimal BB Setup With PA.

Honestly the easier question would be “what CAN’T you run the Beat Buddy through”…

Even though it might be a decent idea to run the BB thru a PA board, consider that it may not sound “Natural” coming from speakers that far apart. It was stated earlier that two small amps (for stereo) might be a better choice and have them sit center stage like a real set of drums is normally set up giving a smaller area of separation. I have had great luck with my guitar amps since I run a stereo pedalboard anyway, and the amps do produce all frequencies well and they give me ample power for any and all pedals and of course my guitar and bass sounds. The Beat Buddy sounds natural, especially when I run it thru a Volume/Tone control box to taylor the sound even better. I like this setup the best and it’s also set up to do direct recording right to a PC where I can adjust all levels as needed.
Just thought I would mention this.
Sincerely, Fingerstyelpicker.

I like using BB through a separate keyboard amp (Roland KC-550 with a subwoofer out to a bass amp underneath it) set directly behind us, right where a drummer would normally be, in duo/trio situations. Sounds a lot like a drummer behind a drum shield, since the snare and cymbals aren’t always killing us on a small stage.
It really feels/sounds very natural, to the point sometimes you almost forget you’re using BB. (Except when BB is late for the gig, or drunk, forgot his stick bag, or in a bad mood because he’s fighting with his girlfriend who is at the gig, also drunk, and trying to make BB jealous by dancing with other guys, after getting in an argument with the club owner about why SHE doesn’t get a discount since she’s “part of the band”… Oh wait that’s not BB, that’s the drummer that BB replaced!!!)LOL
Oh, and did I mention BB can count to 4 IN EQUALLY SPACED INCREMENTS!!!