Is it possible that an aircraft could continue to fly and then land safely on a single engine if another stopped working?
The answer to your question strongly depends on what kind of airplane you're talking about (in particular, what set of rules it was certificated under), as well as the phase of flight you are in (takeoff vs. cruise vs. approach/landing) and whether you are in a jet or a (turbine or piston) propeller aircraft. (I'm going to exclude single-engine airplanes from this discussion, as well.)
In general, an engine failure in cruise flight is a manageable emergency unless you are in extremely high terrain in a light twin or some other corner case. The general procedure is to "drift down" in a slow-ish descent to your single-engine cruise altitude -- if you don't already have a driftdown airspeed worked out, you can use Vyse as a substitute to get a slow driftdown -- and then divert to a nearby suitable airport to get the busted engine looked at, as you'll be flying lower and less fuel-efficiently as a result.
Takeoff is the most troublesome time for an engine failure to happen, bar none: you are at high gross weight and don't have altitude under you to turn into airspeed and landing options. What you want to do is get to the pattern altitude, fly a pattern, and land in a reasonably quick, but orderly fashion -- you don't need to ultra-hurry things even if you have an engine on fire, but you do want to get back on the ground in a reasonable timeframe to get things looked at.
Light Twins (cabin class, commuter, pre-FAR 25 airliners)
These aircraft, unfortunately, struggle in single-engine situations on the best of days, as the certification requirements for single-engine performance in FAR 23 are minimal at best. The common adage you will hear about light twins, even, is "the second engine will carry you to the scene of the crash" -- it is possible for a light twin to be type certificated with nil or negative worst case single-engine climb performance in the case of an engine failure at or shortly after takeoff.
Business Jets (certificated under FAR 23)
Corporate or business-type light jet aircraft (Citations, Gulfstreams, LearJets, Beechjets, etal) and their turboprop counterparts (such as KingAirs), while often certificated under Part 23, will generally have superior single-engine performance to light twins, although this is by no means guaranteed. You'll likely get up into the pattern and be able to make a reasonably normal single-engine landing, barring any aggravating factors at least; this is because even the smallest turbine engines are large enough that the aircraft will have some degree of excess thrust as a twin, leaving it with a thrust reserve in a single engine situation. These aircraft are also flown by pilots with type ratings and a CPL or ATPL, which means they will be more familiar with the performance of that specific aircraft and any aircraft-specific engine-out procedures needed to eke out maximum single-engine climb rates.
Twin-Engine Airliners (transport class twin turboprops and twin jets certified under FAR 25)
The FAR 25 requirements for engine-out takeoff performance are much more stringent than FAR 23, and practically guarantee that the aircraft is capable of positive climb in a V1 cut (engine failure right after decision speed) at maximum takeoff weight. In addition, V1 cuts are a classic airline training scenario that every airline flight crew on the planet runs in the sim on a regular basis (I wouldn't be surprised if a V1 cut came up every few sim sessions or so, if not every sim session).
In a practical sense, this means that unless the aircraft is overloaded or is suffering from something beyond a simple engine failure (I'll talk about that later), an airline flight crew will handle engine failure on takeoff in an airliner as a routine event.
3 and 4 engine airliners
A single engine failure on a three or four engine aircraft is by no means critical, especially on three- and four-engined jet transports. This doesn't mean that 747 crews don't practice V1 cuts any longer, though!
Props vs Jets
Engine failure in a jet, whether it be a T-37 or an A380, is simple: you put appropriate rudder in, ID which engine died (between your instruments and "dead foot, dead engine", you should have enough data to go on), add power as needed on the working engine (say, if you are doing an assumed/FLEX temp takeoff), and kill the dead engine, then run the rest of the checklist at a leisurely pace now that safe flight is more-or-less assured.
Propeller aircraft are a bit more complicated though in that you need to feather the dead engine's propeller to keep it from dragging the airplane down -- older constant-speed prop designs such as those on warbirds and piston-engine airliners had FEATHER buttons for this job, one per engine; newer aircraft with constant-speed props have a gate on either the power or the condition levers that serves in place of the feathering button. You also have more levers to manipulate when adding power to the working engine(s) and killing the dead engine, and may have more difficulty figuring out which engine died without positive flight control responses (I can only imagine how baffling a center engine failure on a Trislander would be!).
As a result of the critical nature of the feathering process, modern (and even some not-so-modern) transport-class (and some other) propeller aircraft have an autofeather or NTS (Negative Torque Sensing) system that will automatically drive the blades of a propeller towards (or even all the way to) the feathered position if it is armed and senses that the propeller is trying to drive the engine. However, feathering the propeller on an engine that isn't shut down is also a bad idea as it causes the propeller and engine to overspeed due to the airloads the engine was driving going away.
Of course, if you are in a light twin with fixed-pitch propellers, none of this feathering talk applies. Be prepared for the unexpected emission of foul language during engine-out procedures, though!
An engine failure during the early phases of approach is generally unproblematic -- it's similar to a failure at cruise, just minus the driftdown. However, an engine failure on short final can be troublesome due to the high workload of an approach, combined with the thrust asymmetry induced during a full-power, single-engine go-around. Prompt rudder inputs are essential in order to maintain aircraft track and control during such a maneuver, and engine failure checklists are best deferred until stable flight.
Engine failures during the landing roll are rare, by the way, and typically a non-event.
Worse than Engine Failure
There are engine damage scenarios that represent a greater hazard to the aircraft than a simple engine failure, or even a fire in an airliner engine. These fall into three categories: uncontained turbine engine failures, engine detachments, and propeller failures.
Uncontained Turbine Engine Failure
Turbine engine failures are divided into two types: "contained" failures where the busted parts fly out the tailpipe, and "uncontained" failures where a turbine or compressor disk basically explodes radially, sending parts flying in a 360° radius around the engine; at this point, further damage is a game of aerial roulette, and can range from a few holes in the cowl to near-catastrophic levels of damage that pose a threat to the entire aircraft.
Engines, even when pod-mounted, rarely fall off airplanes, and much of the time when they do, they depart quietly, a yanked-back thrust lever and nil engine readings the only signs of their absence visible to the flight crew. However, a detached engine is somewhat unpredictable in its behavior, and can strike another engine on the same wing or another part of the aircraft, leading to further damage that can compromise aircraft systems or impair the operation of that neighboring engine to the point where the thrust asymmetry becomes uncontrollable. Nacelle-mounted engines are actually worse in this regard -- a nacelle-mounted engine departing its nacelle leaves behind a big, draggy air scoop that will severely hamper single-engine flying, and on a light twin, this is simply insult atop injury as you have gone from "barely making it to your site of choice" to "we need to pick the nearest open spot NOW because we're trying to fall out of the sky in a hurry"
Propellers do not fail nearly as often as engines do, but when a propeller fails in flight, the results can be catastrophic; an overspeeding prop can fly apart violently or rip its mated engine apart while producing asymmetric forces that have the potential to overwhelm all flight control, while the inability to feather a failed engine's propeller turns it into a drag brake on that side of the aircraft, crippling any attempt to obtain reasonable single-engine climb performance.
Yes. All two-engine (and three- and four-engine) aircraft are certified to safely land if one engine fails at any point on the route, including even the most critical parts of take-off.
However, lower engine performance will force a reduced service ceiling. Basically, the plane will have to fly lower, and less efficiently, so often it wouldn't be able to get to its intended destination and instead would have to abort to a nearby airport.
Short answer is yes, and the immediate challenge is correcting for the unbalanced thrust from the remaining engine. It is certainly a disconcerting feeling flying on one engine when you are used to 2. Kind of like a high wire act without a net.
Sharing a bit of humor here. I flew the A7E a single seat, single engine, light attack jet for the US Navy. We never liked declaring emergencies, and would often use euphemistic language in our communications over such small affairs as engine troubles. I heard this story a while back and thought it apropos to the question.
A military pilot called for a priority landing because his single-engine jet fighter was running "a bit peaked." Air Traffic Control told the fighter pilot that he was number two, behind a B-52 that had one engine shut down. "Ah," the fighter pilot remarked, "The dreaded seven-engine approach."