You can make a lamp out of anything that you can keep burning. Before there were kerosene lamps, there were candles and before that, there were all sorts of oil lamps. Before that, there were torches. They all had the same purpose: To give light.
You can use the same simple, basic principle to make oil lamps out of almost anything. Any kind of vegetable oil, liquid or solid, and any kind of animal based oil, will burn.
A wick is anything that will "wick" or absorb and draw up the oil. Braided cotton or cotton twine is commonly used, but thin pieces of wood, twists of rags or even twisted dry grass will work if properly prepared. To prepare a wick, it has to be soaked in whatever you intend to burn.
A wax (animal and/or vegetable "oil") candle needs a wick that has been soaked in that wax and allowed to harden. To make a wick for a dish of, say, fish oil, soak it in the oil before positioning to burn, and so on.
To keep a wick up out of a container of liquid fuel, you will need a wick holder. A kerosene lamp is an example of how you might make one, but it's very simple to use a piece of wire. Make a flat spiral to set on the bottom of the container, then a piece coming up from the center of the container that is higher than the fuel level will be. At the top of that, make a round "eye" to fit the wick through so it won't fall into the fuel.
Soak the wick then thread it through the eye and let it fall to the bottom of the container. Fill the container with fuel and light it... and there you have it.
A few of these around a room will light it as well as candles. It's a good trick to know when the lights go out and you're not prepared.
A second way to make a light and one that you don't have to mess with much at all, is to use a can of shortening or lard. Insert an ice pick or something similar, into the center of it, then drop a wick (which you've soaked in a liquid version of the solid fat) into the hole and tamp it down into it. This will burn for hours and hours.
A good survival item is a can of tuna in oil. You can eat the tuna, of course, then you can make an oil lamp of what's left, using the label (soaked in the oil) as a wick.