Are there other life giving sources of energy in space apart from stars? (like nebulae, radiation, etc.)? Or are all possible life giving potential sources some variation of a star?
There are three important sources of energy in planetary bodies (or their moons) that may be important in determining their suitability for life. All of these are evident in various bodies in the Solar System.
The first is tidal heating. When one object orbits another then there will be gradient in the gravitational force across their finite sizes. If this is combined with any orbital non-circularity or difference between rotational and orbital periods, along with some fluidity in the structure of one of the bodies, then it will be heated by frictional forces as the fluid moves in response to the changing gravitational gradient. This heating is most evident on Io in our own Solar System, the closest large moon of Jupiter, and is generally more important for objects orbiting close to more massive stars or planets.
The second is heating by radioactive decay. When planetary systems are born, they can incorporate significant amounts of radioactive material. Some of this has a short half-life and may be responsible for a short period of intense heating, but other isotopes are much longer lived. According to this article, about 20 TW of power is generated inside the Earth from the radioactive decay of uranium, thorium and potassium. This sounds a lot, but is about a factor $10^4$ less power than we receive from the Sun.
The third source is just the "heat of formation". Gathering together the raw materials of a planet/moon and compressing it into a gravitational potential well, inevitably leads to the generation of considerable heat and it can take billions of years for this heat to leak out into space. Indeed, for the Earth, the heat of formation is probably comparable with radioactive decay in terms of supplying energy to the surface from the interior.
Stars function by nuclear fusion. There is energy that isn't released by nuclear fusion, nor from the nuclear decay of elements produced by fusion.
There is the cosmic microwave background. This is radiation, but it "cold", at about -268 *C. There is a lot of energy, but because it is so "cold" it is difficult to use it to power anything.
There is a lot of gravitational potential energy. This can be released when things fall together. So when an asteroid hits the Earth, the energy released is not from a star. However its not a dependable source of energy for life.
There are proto-stars that are powered by gravitational energy, and there are accretion discs around neutron stars and black holes that are hot due to the gravitational energy released as matter falls into them. The hard X-rays from accretion discs isn't much good for life.
There's also a lot of potential energy in a cloud of interstellar gas that could be released if it all fused to heavier elements. However the only effective way to release this energy is to make stars: Lots of energy, not easy to get at.
In principle, all mass is energy, by E=mc². However, using this energy to power something isn't easy.
Really you are asking the wrong question. There's lots of energy. The trouble is converting it to do useful work, otherwise all this energy is about as useful as a piece of coal to a hungry person.
ProfRob makes a good point about tidal heating, but you can have internal heat sufficient for life even without this. The heat I'm referring to is heat trapped in a planet's core from its formation, which slowly and reliably leaks out through geothermal vents.
We observe this at the bottoms of Earth's oceans, where thriving colonies of deep sea organisms live in total darkness, feeding off the heat and chemicals that come from these vents. No sunlight supports these organisms, only geothermal energy.
Note that how long this is useful depends on your planet's size. Mars used to have a liquid core, but has since cooled down enough to have its core freeze since it's smaller and therefore cools faster. The larger your planet, the longer it'd be able to provide this trickle of heat.
Edit: it has been made aware to me that Earth's internal heat is mostly driven by radioactive decay nowadays, which I didn't know. The point still stands however that a planet's original heat could be sufficient to jumpstart life. I'm not sure how long it could provide for this before something else would need to kick in though.
You can also have planets with powerful magnetic fields, a conductive body passing through them generates electric currents, that besides simply heating can also power chemical reactions directly (like charging up a chemical battery).
There is also chemical energy. If there is a constant supply or accumulated store of reactive chemicals, that can act as fuel or food or battery electrolytes. For example, there are clouds of organic molecules in space that could rain down on to a planet and be used as 'food', or which could collide with other clouds and react. Or planets made of different chemicals could collide and then react with one another.
As rotating dust clouds collapse under gravity, they can form accretion disks in which stars and planets form. The continual collisions generate heat, causing them to glow in the infrared. This likely includes cases too small to form a star (brown dwarf systems).
Shock waves passing through interstellar gas clouds can produce intense heating. These shock waves are usually produced by stellar explosions (supernovas) but could in principle be produced by anything that causes high speeds, like the bow shock of a planet passing through a gas cloud.
The answer by mentioned two internal sources of energy in planets and moons, etc.:
one) Tidal heating. In some situations tidal heating hasno significant effect, in ohters it can make a cold moon warm enough for life, in others it could cause a runaway greenhouse effect.
Two) Energy produced by the decay of radioactive isotropes.
But there is at least one other sourceo of internal energy in planets and moons, etc.
Three) The left over heat from the formation of the world as grains of dust, molecules of gas, and larger objects fell thousands or millions of miles toward each other and eventually impacted with great force.
There are also electric bacteria. The BBC's There are microbes that eat and poo nothing but electricity coverts the topic extensively.
At first electrical forms of energy absorption and excretion seemed exotic, but this now is seen as a widespread phenomenon on Earth, including colonies that include multiple species working together to literally complete the circuit.
These organisms don't derive their energy directly from Sunlight or heat, but the ones currently found on Earth do require enough warmth to have liquid water environments in which to ply their trade.
So perhaps as long as self gravitation leftover heat and radioactivity mentioned in other answers can provide enough warmth for a liquid water environment, bacteria-like life can feed directly off of the mixtures of primordial chemicals (organics, metals, oxides, sulfides, etc.) without necessarily eating the chemicals themselves, but instead forming electrical pathways through which electrochemical reactions can take place.
From the BBC article:
While most organisms get their electron fix from carbohydrates, some bacteria can harvest electrons in their purest form. They can effectively "eat" electrons from minerals and rocks. In a way, they are getting their electrical energy straight from the socket.
Annette Rowe, a graduate student of Nealson, has found six new bacterial species on the ocean floor that can live off electricity alone. All are very different to one another, and none of them is anything like Shewanella or Geobacter.
She found that, when no other food source was available, the bacteria would happily take electrons directly from the electrodes. In their natural habitat, the bacteria likely take their electrons directly from iron and sulphur in the seabed.
Examples of electron-eating bacteria found by Rowe include Halomonas, Idiomarina, Marinobacter, and Pseudomonas of the Gammaproteobacteria, and Thalassospira and Thioclava from the Alphaproteobacteria. Many more electron-loving bacteria have now been found. In fact all you have to do is stick an electrode in the ground and pass electrons down it, and soon the electrode will be coated with feeding bacteria. Experiments show that these bacteria essentially eat or excrete electricity.
Some species of Geobacter, he says, can both directly transfer electrons to electrodes and also directly accept electrons from them. In 2015, we learned that electron-eating and electron-excreting microbes can actually team up and pass electrons between each other, wiring themselves into a common electrical grid.
It turns out multiple types of bacteria can work together:
Different species of bacteria and archaea – ancient single-celled microbes similar to bacteria in many ways – team up to degrade the methane before it can get the surface.
Gunter Wegener from the Max Planck Institute for Marine Microbiology in Bremen wondered how the process worked. He collected samples of the microbes, which live at temperatures of 60C on the ocean floor, and put them under a scanning electron microscope.
The microscope revealed thin wire-like structures protruding from the bacterial cells. Although only a few nanometres wide, the wires were several micrometres long, which is much longer than the cells themselves. It seems that the bacteria use these nanowires to hook up with the archaea.
The archaea feed on electrons from methane, oxidising the gas to generate carbonate. They then pass the electrons on to their partner bacteria along the nanowires, which act like power cables. Finally the bacteria deposit the electrons onto sulphate, producing energy that the cell can use in the process.