My battery charger for my boat can charge at either 2, 10, or 15 amps. What is the difference in these and which should I be charging my battery with?

I am not an expert, but as I understand it using 2 amps, although it takes  longer, provides a better charge to the battery. Of course if you need the battery charged quickly use the higher amps. Also, charge the batteries as soon as possible after each use.

Charging at 2 amps is not necessarily better and if it is a lead acid battery, it may be worse.  For most modern chargers with automatic control, the 10 amp setting will be the standard.  The 2 amp setting is for long term storage maintenance and the 15 amp if for emergency recovery however most chargers can't maintain their maximum output for long.

These settings are a maximum current.  Setting it at 10 or 15 amps does not force that amount of current into the battery.  The current will taper off to zero as the battery approaches full charge.

The newer 'Smart Battery Chargers' will charge the battery at 30-40 amps until the batt comes up to State of Charge (soc) of %75. They call this a bulk charge and it is determined by the voltage across the battery terminals. They then fall back to a 2-5 amp charge rate for the remainder of the charge cycle. Their claim is that the lower final charge rate allows the battery to accept a greater final charge level than the  higher amperage would permit.  They will also dump up to 80 amps into a completely dishcarged battery for 5 mins to permit a vehicle to be started. So apparently the current isn't an issue until the battery reaches a 75% soc.

My antique charger charges at a 10 amp rate until the battery is fully charged and then falls back to a trickle charge until it is turned off.

So the short answer is go ahead and charge at 10 or 15 amps, and if you want maximum energy in the battery, drop back to 2-5 amps for the last few hours of charging.

Incidentally a company called Firefly Energies has developed a carbon/lead battery with 2-3 times the energy storage capacity of the best lead/acid batterys now available. They claim their test folks have charged it a 300 amp rate to full charge. They also claim it will provide 800 discharge/charge cycles with no significant loss of capacity. Both of those numbers are at least 3 times better than todays lead/acid batterys. These batterys are compatible with todays manufacturing processes, and therefor should be comparably priced. They plan to be in production this fall.