K0: A K0 civilization is grounded to a single planetary body, and has very limited means of energy production. Examples include any society found before Neolithic times.
K0.5: A K0.5 civilization is often grounded to a single body, but may have ventured out a little into the void of space. This is where all human civilizations have been since the beginning of the Classical Age. During this period, a civilization begins to transition to a K1 civilization, and gains the ability to either commit species-wide suicide or make sure that its species becomes nearly immortal. This is the most dangerous time for a species.
K1: A K1 civilization has begun to reach out into the cosmos in earnest. At this point, a civilization will likely have colonized its entire solar system, and parts of nearby star systems. At this point, a civilization consumes the total energy output of an entire planet. By this point, a civilization has usually gone green. Good examples include the worlds of Buck Rogers and Flash Gordon. These civilizations are often seen in 50s and 60s optimistic imaginings of our future.
K1.5: Civilizations on this level usually create small interstellar nations and make contact with similarly-advanced alien races. Picture the Federation from the original Star Trek series.
K2: By this point, a civilization consumes a level of energy that can only be naturally produced by an entire mid-sequence star. K2 civilizations can perform truly awesome feats of physical engineering, such as building Dyson spheres and other megastructures. A perfect example is the Federation from Star Trek: The Next Generation. K2 civilizations are usually interstellar nations, and almost always have FTL capabilities.
K2.5: These civilizations often cover hundreds or thousands of star systems. A good example would be the Kree Empire from Marvel Comics, the Forerunners from Halo, the Invaders from Independence Day, or The Dominion from Star Trek: Deep Space Nine. These are not to be messed around with.
K3: This is where I make you feel like a tiny, inconsequential blink of the eye in a vast, vast universe that doesn't care about you. K3 civilizations consume enough energy that they must harness an energy source on par with the natural energy output of an entire galaxy. That's huge. The Milky Way is an average galaxy, and it holds between 100 and 400... billion stars. That's between 100 and 400 billion times as much energy as a K2 civilization consumes. It's hard to wrap your mind around. The nearest examples in pop culture would be the Precursors from Halo, the Galactic Empire from Star Wars, and the Borg from Star Trek: The Next Generation and Voyager.
K3.5: Bigger. Better. Scarier. Think Species 8472 from Star Trek: Voyager. Those guys were able to bring the goddamn Borg to their knees and tossed around the idea of wiping out all life in the Milky Way like it was something they'd do if they were bored on a Saturday.
K4: Hey, I heard you like realism. Too bad. You thought the K3 civs were scary? Well, these guys consume the energy of an entire universe! That could potentially be trillions of times more energy than a K3 civ. I literally don't have any examples here, so the only thing I'll say is that the Christian God is just a single member of a K4 civ.
K4.5: These guys pee on universes because they can. 'Nuff said.

S0: About where humanity was before 1940- very, very little computing power, and at first we had none other than the brains of mathematicians and their abacuses.
S1: This is where humanity is right now- on the verge of breaking Moore's law because our chips are getting to small to contain heat leakage. In other words, quantum computing, molecular computing, and bio-computing is feasible within 50 years of an S1 civ.
S2: Quantum computing, bio-computing, and molecular computing are now widespread and are making massive leaps in computing technology.
S3: Computers are now reaching human-level AGI.
S4: This is the Turing Barrier- beyond here, AGI becomes ASI, and computers rapidly become smarter than humans in every aspect. Computers are now, without a doubt, sapient.
S5: Computers in an S5 civilization now no longer meet the definition of computers- they are fully-fledged consciousnesses that can do unimaginable things by today's standards- things our human brains do not have the capacity to comprehend.

SP0: SP0 civs only take up a small part of a planetary body.
SP1: SP1 civilizations take up an entire planet, plus parts of other heavenly bodies.
SP2: By now, a civilization has gone interstellar and has colonies in other star systems.
SP3: Beyond this point, a civilization is so expansive, it is easily visible on a galactic map.
SP4: SP4 civs control an entire galaxy.
SP5: Civilizations on this level control multiple galaxies.
SP6: These civs are rare, controlling entire universes. These are the largest civilizations imaginable.

TC0: Travel is limited to travel by foot or using domesticated animals or boats.
TC1: Mechanized travel lets organisms travel across larger distances- steamboats, cars, trains, and planes are all developed at this point.
TC2: Space travel is now a fully feasible thing. At this point, nearby heavenly bodies, such as moons and close-by planets, are landed on.
TC3: Travel across a species' home star system is now easily possible with speeds approaching or exceeding 0.25 c.
TC4: Travel between stars is now easily possible with travel approaching or exceeding 1c./space-time can now be bent, allowing for travel through wormholes.
TC5: Galactic travel is easily possible, and crafts move at speeds exceeding 1,000c.
TC6: Intergalactic travel is now possible and common. Starcraft can travel at speeds of 1 million c or greater.
TC7: This is the ultimate end-stage of travel. Travel between universes and through time is possible, and ships can instantly transport themselves across distances measured in billions of lightyears.