As others have noted, the purpose of boost pumps is several; boost pumps move fuel, they supply head pressure to reduce or eliminate vapor lock, they ensure a steady supply of fuel to higher-pressure pumps, they refill carburetor bowls, and in some cases (not all) they serve as backup supplies to the engine driven pump.
In many cases, boost pumps will not suffice to provide enough fuel flow to the engine at high power settings. It's for this reason that sometimes when a pump failure is suspected (or verified) one can operate at reduced power.
Lack of pressure to the engine driven pump can result in pump damage. It should be noted that where an engine driven pump fails, absent a shaft or drive failure, it may serve to damage or contaminate the accessory section of the engine. An engine shutdown may be necessary, or even eminent.
Boost pumps may have sufficient flow and pressure (two very different things) to supply the engine, or they may not. Auxiliary fuel pumps come in many flavors, and applications. In one fuel system, pumps may be used for nothing more than transferring fuel from one part of a tank to another, to ensure fuel flow or availability at various flight attitudes. Other pumps may exist to supply the enigne, or other tanks, and some pumps may be boost-override pumps that provide higher pressures than others, to ensure that the boost override pumps can supply fuel in the event other pumps are not available. Some pumps may be used for little more than to dump fuel (in advanced aircraft). In some aircraft, the engine driven pumps require fuel for cooling and lubrication, and need a boost supply to prevent cavitation or EDP damage.
In some cases, the EDP is necessary to provide the fuel flow necessary to activate certain engine controls, or the fuel control unit. Boost pumps in some cases do not provide adequate fuel pressure (or flow) to handle this, as well as to supply the engine.
Some aircraft use high and low boost positions, for various reasons. When you turn on your boost during an emergency procedure, you may not be doing so to supply the engine, but to restore flow to the engine driven pump or to other parts of the fuel system.
The Cessna 200 series of piston aircraft utilize a small header tank located at the base of the wing struts (or where the struts would terminate, in the case of the 210); these tanks receive fuel from the wings, and send fuel to the engine driven pump, fuel control, etc. They're important, because they're not just a point through which fuel passes when headed to the engine. They're a point where bypass fuel off the pump is returned, and that leads to a critical issue called "fuel flow fluctuation."
Cessna has a procedure for FFF, which involves applying boost pressure, adjusting mixture, and switching tanks. This procedure has been in the Cessna manuals for several decades, and it's dead wrong. In fact, it can assure that you don't get the engine restarted. A little systems knowledge will bear that out.
When the engine stops during fuel flow fluctuation, it's often because hot bypass fuel, returned to the header tank, is trying to climb back up the supply line from the wing tank. The result is stoppage of fuel flow, vapor lock in the feed line from the wing tank to the header tank, and an engine failure or intermittent power and fluctuation. Applying boost pressure cements the deal; it ensures that additional fuel is flowing to the pump, increasing the amount of bypass sent back to the header tank, making the problem worse, and often making a restart impossible.
The key to understanding what to do is knowing the system; the header tank feeds the fuel selector before going elsewhere. Rather than turning on the boost pump, reduce throttle and switch tanks. This situation doesn't occur often, but when I've experienced it, swapping tanks did the trick right away. Restoration of an undisturbed cool fuel supply fixed the problem. Then and only then could I concern myself with boost, adjusing the throttle and mixture, and referring to a checklist. Know your aircraft and what the various inputs do.
Ever applied high boost in flight and watched its effect on fuel flow, engine operation, CHT's, EGT's, etc? In some systems, it's possible to kill the engine with boost, or cause it to load up, and even lead up.
Even in the same type aircraft, significant variations in the fuel system (and other systems) can exist. I operated a certain type of piston twin in a hostile overseas location that had some very significant differences in the fuel system from civil-use counterparts. Those systems gave us up to a 12 hour loiter time at reduced power settings, but like many piston twins, also meant that under certain circumstances, not only would a portion of the fuel be unavailable, but the useable fuel might be bypassed and transferred to an unusable location. Not all pumps on board fed the engine, and not all pumps would run the engine, and when the engine was running, it might be crossfed from an unconventional tank, but would bypass fuel back to a conventional location...possibly making it unavailable later on.
Know your system, but know also that there are some aspects of operation that you may not learn until the actual failure of that engine driven pump, or other system failure.