Space heaters are a nice way to heat up an individual room in your house, but also a relatively dangerous way. Stoves and ovens cause more house fires, yes, but nearly everyone has a stove and most people use it regularly. Space heaters come in #2 despite fewer people having them, and them sitting unused most of the year.
Space heater are pretty simple: they take a lot of non-heat energy and turn it into heat energy.
Heat
Your space heater can put out enough heat to ignite your flammable curtains or your flammable bed. Keep cloth and paper away from your space heater. (The recommendation is generally 3 feet, or 1 meter)
Energy - Non-electric
If non-electric, your space heater is burning something. This burning is hopefully contained, but burning things also releases gases, such as carbon monoxide. If you're using a non-electric space heater indoors, you should have it connected to the outdoors to ventilate. It's also not a bad idea to have a carbon monoxide detector.
Energy - electric
If electric, the non-heat energy causes fires the same way it heats your room: pushing a lot of electricity through something that doesn't like that much electricity turns that electricity into heat. If that heat's spread out where you want it, great! If it's concentrated in the wrong place—such as your house's wiring—ungreat.
What doesn't like lots of electricity? Long things and thin things. You should always plug your space heater into the outlet directly, not an extension cord or power strip. (Would a short, thick extension cord kill you? I don't know. Better not risk it.)
Additionally, nothing else should be plugged into your outlet. In the US, appliances can't use more than 1500 watts (except in short bursts) for safety reasons, and space heaters almost always use exactly that much. If you're running a hair dryer from the same outlet, that's too much power and you'll trip your circuit breaker if you're lucky and start a fire if you're not. Ideally, you'd have a dedicated circuit for your space heater.(My house has dedicated circuits for the window A/Cs, and we just use those same outlets for space heaters, since no one uses both at the same time. I hope.)
Note that I said your space heater uses 1500W. If you're in the US, your "small room" electric heater likely uses 1500W, same as a "large room" space heater. Some have low-power settings (usually 750W) as well, but manufacturing one that uses 1500W costs about the same as a 750W one, so most are just 1500W. Also, space heaters are 100% efficient, so 1500W usage = 1500W of heat. Room size labels are mostly just marketing.
There are various forms of electric space heaters, and they might distribute heat differently. If you've seen those concave metal dishes look like fans, they're directing IR radiation right at you. You'll usually feel this heat almost immediately.
Others may heat up a thermal mass, often oil. This will take longer for you to feel the heat, but it's the same amount of heat going into the room.
Other Safety Stuff
The features that make a space heater safer than others are tip-over detection (especially useful around kids and pets), overheat protection (to auto-shutoff if it's too darn hot), and cool-touch housing (also nice for kids). Additionally, UL certification means a third-party had approved its construction/design as "safe enough".
People always say to put space heaters on flat surfaces, and specifically on the floor, not a shelf or table. I believe this is because you don't want it falling and no longer being upright. Also, it falling on someone would be bad.
Additionally, it's advised to never leave a space heater on unattended (including "you're asleep"). I'm sure my housemates ignore this rule, and I suspect you will some nights as well. Just know that you've been warned and test your smoke detector.