I sit on my city's planning board; it's a related group that hears more major (typically coming from commercial development) reviews, but the process is really similar.

There will be an applicant presentation, a public comment session, and the board will typically question the applicant. As a general rule, if they're at this point, they're looking for an exception from a standard rule so that they can avoid either a setback requirement, a wetland requirement, or something to do with historic district rules. Setbacks and wetlands are pretty common where I am, historic stuff has its own board, but those can vary by location. I would stronglly suggest that you talk to your neighbor about concerns before you go to the board; in a lot of cases, neighbor fights erupting at these meetings could have been resolved by people actually talking to each other beforehand.

If you have an issue with what your neighbor's doing and they're unwilling to address it with you, come to the meeting prepared with the specific ordnance or rule they're looking for an exception from and a reason based on something more than 'I don't like it'. We had a recent case where multiple neighbors were objecting to the noise of earthmoving equipment being used on a subdivision project. I could sympathize - I grew up in Nashua during a major construction boom, and I got to contend with the noise of dump trucks and equipment moving up and down my street while I tried to do homework - but if they're working during times that aren't set as quiet by ordnance, they're allowed, and a ZBA or planning board that denies for that is going to get sued and lose, which means that your neighbors are going to get to do what they wanted and your taxes are going up to pay for the legal fees.

If you do have objections, I'd recommend practicing your presentation beforehand. The first couple times you're speaking on camera with an at least partially hostile audience are surprisingly intimidating, and you'll do better if you're prepped beforehand. Good luck!