So you’re planning your spaceship or spacestation (which, frankly, are much the same thing) and you decide that artificial gravity sounds like a good idea, after all human bodies are evolved to survive with a sense of down. It makes things like eating and going to the toilet a lot more dignified. Lack of gravity also creates some long term health problems which come from the degradation of muscle mass and calcium density in bones which eventually lead to kidney failure. Unfortunately the anti gravity drives of Sci Fi films are fictional and wearing some magnetic shoes to stick to the floor doesn't help with none magnetic stuff, such as all that food that you are planning to eat. NASA make heavy use of velcro, but there is a catch all solution.
By having a ship designed like a cylinder and spinning it, we can live on the inside and there will be a resultant force pushing us (and everything else) to the edge. The exact speed it would have to spin at would depend on the radius of the ship, but the general rule is that the smaller the radius, the larger the angular velocity. However there is a payoff here; for smaller radius ships the acceleration acting on the human would be large enough to kill us, while larger ships are harder to get into space in the first place, let alone provide them with enough kinetic energy to get them spinning.
The king of rotation ships is Arthur C Clarke, who not only included the famous rotating space station in 2001: A Space Odyssey (he wrote the screenplay for the film with Stanley Kubrick along with the novel), but he also wrote the novel Rendezvous with Rama in 1973 which deals entirely with a group of humans exploring a huge alien rotation ship and focuses on the weird physics involved.
For instance the crew enter a huge alien ship, which they designate Rama, in the centre of one end. They then climb a ladder which takes them to the rim which is about 10 km down (down as in the direction of the resultant force, not down as in towards the centre of rotation, although you happen to be standing on something where both of those meanings indicate basically the same direction). As they go the force they are under increases linearly as they get closer to the bottom. So at the centre you are very close to being weightless, but any deviation from being exactly on the axis of rotation would very slowly drag you further away. As you went the force would increase and you would fall quicker and quicker until you hit full acceleration just before impacting rather fatally with the rim.
If you climbed the ladder from one side of the rim, across a diameter, to the other side, then what would feel like the natural orientation to climb in (i.e. with your head above your feet) would end up with you upside down as you got to the other side.
Rama is depicted as being about 50 km long and so is big enough to have localised weather. Half way along the ship is a lake which goes the whole way around the rim. This means that if you are in a boat in the lake you can look directly up and see part of the some lake suspended above you. Something about this seems paradoxical.
Here is a beautiful image which someone created with the same idea. Whole landscapes curve upwards in front of you and the sky would be filled with more land.
Also worth a mention are the Ringworlds of the video game series Halo which were based on The Orbital from Iain M Banks' Culture Universe and which are rings with large radii, but which form only a narrow band. As they spin you also get a day night cycle from the local star.
How fast would you have to spin something to make the surface gravity equal to that of Earth? In circular motion the radial acceleration (i.e. the acceleration going away from the centre rather than perpendicular to it) = r/v^2 where r is the radius and v is your tangential speed. So for Rama with a radius of 10^4 m we want to get a = 9.8 which is the gravity at sea level on Earth which we call g (although some (well Asimov, but that's enough for me) have argued that the best level of gravity for humans is actually somewhere around 0.8g). This lets us work out v, so v^2 is about 1000, so v is about 31 m/s. To get into RPM (revolutions per minute) we do RPM = v*60/2*π*r = 0.03 RPM which ia tiny.
However if we wanted to get the same level of artificial gravity in the tight radius of the 2001 ship pictured above, the radius would be about 10 m instead, which would give a velocity of sqrt(10/g) = about 1 m/s. This is a much smaller speed, but you will feel it more because you are in such a tight circle. The maths works out at 0.95 RPM which is a lot faster. Humans are ok living below 2 RPM for a short time, but by 6 RPM there are various inner ear problems which arise. Studies by NASA show that a radius of 60 m is about the smallest humans could thrive in; but that puts the size at bigger than we have been able to get into space in one piece so far.
But overall the problem of creating artificial gravity is one of the major obstacles to creating generation ships; ones where more than one generation of human occurs while the ship is on its voyage through space. Given the distance to even the nearest neighbouring star, this is a problem we need to solve to be an extra-solar species. For now there exists only one spinning, generation ship travelling through space and we call it Earth.