Many photographers will tell you that you should always use a tripod for best results. But what if I have the shutter speed set at 1/250 of a second? Just how much would the camera move or shake if I stand still on the ground, legs apart, and hold the camera firmly in my hands with elbows against my body? If we assume that the camera moves 1 mm just as I take the shot, how much would this, really affect the resulting image? Aren't they all just being over-dramatic?
Camera motion is usually measured in terms of angular size or arc: that is, how many degrees, minutes, seconds of arc the optical axis moves during the time the shutter is open. How much the same amount of motion affects the image is also determined by the angular size of the Field of View (FoV) yielded by a particular focal length and film/sensor size.
For the example in your question: At 1/250 second using good handheld techniques the amount of blur will depend upon the FoV. If you are using a wide angle lens such as one with an 18mm focal length, there will probably be very little detectable blur. If, on the other hand, you use a 600mm focal length there will probably be significant blur, especially if you are using an APS-C camera that reduces the FoV to the equivalent of a 900mm or so lens mounted on a 35mm format camera. The reason for this is that the same objects at the same distance from the camera cover over 1,000 times as many pixels using a 600mm lens as they do using an 18mm lens! (33.33*33.33=1,111.11)
Back in the days of 35mm film, the general rule of thumb was to use a shutter speed no slower than 1/focal length when hand holding the camera. This assumed a print display size of 8x10 inches viewed at a distance of 10 inches by a person with 20/20 vision. With digital sensors that are typically smaller, the focal length multiplier must be applied as well (1.5x for Nikon APS-C, 1.6x for Canon APS-C cameras). This still assumes the same viewing conditions, however. If you're going to view sections of the image at 100% on your computer monitor, then even the slightest blur that isn't detectable in an 8x10 at 10" will be evident.
The excellent lensrentals.com blog had an interesting article about this not so long ago.
Linked in the article is an Android app to measure just how much you shake the camera while holding it. Try it out for yourself.
Some of their results are (emphasis and formatting mine):
What does this all mean for your question
The longer the focal length, the more will angular movement (that's the movement relevant here) have an affect. If you rotate by one degree during the shutter opening time, the movement will be half as bad as rotating two degrees. Obvious, isn't it?
The 1/focal_length rule comes in here, because we first have to convert the focal length into viewed-angle through the lens.
The longer you open the shutter, the more time will your camera have to move. Here, a twice as long shutter period will not cause a blur twice as bad. To realise this, look at the chart shown in the linked blog. The Frequency-Amplitude curve is decaying with approximately a negative exponential. But as a first approximation, it is probably fine to say double shutter time means double the blur.
You can also learn to keep particularly still. Many photographers borrow ideas from military snipers. There is some technique involved on how to best hold your camera and then some more technique on how to breath and when to press the shutter button during your breathing cycle.
It depends on the lens and camera you are using in addition to other factors. If you are using a wide-angle lens then there would be no blur but if you're using a telephoto lens having focal length greater than 250 mm (provided there is no image stabilisation on the lens) then the shutter speed of 1/250 will most likely produce a blurred image. In addition to telephoto lens camera also plays a significant role in this , if you're using a APS-C sensor sized camera then due to the crop factor , the resultant focal length will be original focal length * crop factor , for example, for canon aps-c cameras crop factor is 1.6x so a 250mm focal length will become 400 mm and therefore on shooting with 400mm focal length at 1/250 s the image will most likely be blurred (as defined in 1/focal length rule).
Image Stabilisation also plays a significant role in this , some lenses provide image stabilisation (vibration reduction for nikon) upto 4 stops , so upto 4 less stops of shutter speed can be used to shoot the image , but it is still risky . So , it is best to shoot with a tripod as much as you can
Other factors are also there such how steady you can keep your hands which vary from people to people and which also depends on the weight of lens (as telephoto lens are usually heavy) and telephoto lens are more prone to this as slight angular variation can cause blurred images.
It really depends on several factors including the focal length and how steady you are. If you are very steady, it is fully possible to get sharp photos by hand at shorter focal length even down to the 1/60 or 1/30 range, particularly if you have an image stabilizing lens. At longer focal lengths with less steady hands, it can become a problem even around 1/250.
The problem isn't distance traveled (which only the most recent built in image stabilization technologies even try to deal with), but rather angular changes in the direction the camera is facing. We are pretty good at holding something still in terms of position, but far less so at keeping it pointed in the same direction. This is mostly because of how low tolerances are. When you are shooting a photo at super-telephoto lengths, very, very small changes in angle of the lens cause radical shifts in the image.
Optical image stabilization attempts to counter for this by using a floating lens element that is stabilized by gyros to maintain it's orientation and compensate for changes in angle of the lens, but there is still a limit to how much they can do before either a) introduces a noticeable and unacceptable lag in intentional panning, b) an unacceptable amount of weight and power consumption or c) simply goes outside the area that the lens is able to see and thus looks at the inside of the lens instead of the outside world.
Tripods are certainly the most steady, but monopods have a major impact on stability as well as it prevents you from having to support the full weight of the camera and gives a third point of stability (your two feet and the monopod). I'm pretty steady, but shooting with a monopod, I can shoot 1/60 second shots at 200mm without issue while using the optical image stabilizer. Doing that handheld would require considerable focus in addition to very steady hands.
Without any support, the 1/focal length rule is a decent guideline, but there can be a lot of variation from person to person (I've heard a range of 1/focal length to 1/(5*focal length)), so experimenting with your capabilities is still the best bet. But the general answer is no, you don't need to use a tripod for everything. It often won't even significantly improve your shot as long as blur isn't an issue for the shots you are shooting. Shooting 1/250, I'd rarely bother with a tripod unless I needed it for something like a panorama.
I use my Point-and-Shoot[PNS] camera at 1/60th of a sec with not much blur. I start getting blur at over 1/40th to 1/30th secs. I can post a few photos as proof if what I said seems totally bonkers.
Edit: PNS is point-n-shoot. Nothing official, just a way of saying it. And regarding people being over-dramatic about sharpness and stuff ...
"There is nothing worse than a sharp image of a fuzzy concept." a great man once said :)
It's not an issue at 1/60 or faster, but at slower exposures it can be, and even at faster ones if there are high-contrast points in the scene, like small sources of light against a dark background, which leave obvious motion trails that detract from their shape.
When taking a freehand shot using the trigger on the camera, the main problem is that the button press and subsequent recoil add motion to the camera just before the moment of image acquisition.
To reduce trigger-induced camera movement during exposure when taking a freehand shot, without a tripod and external trigger, I use a short self-timer. The two second delay between the button push and image acquisition allows the camera to settle down to its regular background motion caused by the trembling of your hands.
You can reduce this background motion by keeping your wrists rigid, while remaining as relaxed as possible. The idea is that your hand, camera and forearm constitute a single, more or less rigid pendulum. This pendulum is fairly massive, and so it oscillates with a fairly low frequency. You only have to exercise maximal control over unwanted motion during the actual moment of exposure. By counting down along with the self timer, you know exactly when that moment comes.
With practice you can take decent low-light pictures down to 1/20 or so. Slower speeds than around 1/20 require stabilization.