Op-amps (operational amplifiers, that is) typically provide lots of voltage gain but due to a number of engineering constraints they invariably have wimpy output stages that can't source or sink a lot of current - 10 to 50mA max is the usual range. This is where a buffer steps in: they don't provide voltage gain but they do provide lots of current gain, freeing the op-amp from the onerous task of supplying high power.
How many buffers one "needs" to drive a particular pair of headphones is the subject of considerable (and usually non-sensical) debate. If you have low impedance cans (less than 64 ohms, say) then you need lots of current gain but not so much voltage gain. If you have moderate impedance cans - around 100 ohms - then you need a little of both voltage and current gain, while high impedance cans, like the Sennheiser HD-600 at 300 ohms, require lots of voltage gain but very little or no current gain.
Theoretically, one BUF634 buffer should be more than adequate to supply all the output current any pair of cans could possibly take without frying, but some argue that paralleling (or "stacking" as they like to call it 'round these parts) multiple buffers is beneficial anyway. The argument that it lowers the output impedance is a bunch of bs because the buffers are invariably enclosed in the op-amp feedback loop so the effective output impedance will be in the milliohm range, anyway. Stacking buffers does reduce the rise in low frequency distortion that occurs from thermal feedback on the silicon die from the output transistors to the drivers. Most modern buffers take this into account (using Vbe cross-cancellation techniques, typically) so this may or may not be a problem with a particular buffer. Generally, if you feel compelled to stack buffers, then you probably need to use a different buffer or else look into a discrete current gain stage - stacking is just so inelegant.
To be fair, my take on this is distinctly in the minority here, and YMMV anyway.
Can you parallel op-amps (with the obligatory current-sharing resistors!) instead of using a buffer? Well, to an extent, yes, but keep in mind that one BUF634 can provide 250mA of output current which would take a minimum of 5 op-amps in parallel to duplicate. The economic advantage of this technique is rendered moot if the op-amp you are paralleling is an AD8620!