Most of the time we will want to try to use the lowest ISO setting possible to get a good "clean" image, with minimal noise and rich vibrant colors. This type of image is generally sought after in studio and portrait photography, and many lifestyle images also require that clean, simple look - any images where we don't want the added distraction of noise and have colours that are vivid and true to life.
However, occasionally it can be interesting to go the other way and deliberately take images with a high ISO to GET that noise and dulled colors. Noise can add a certain moodiness to an image, a sense of mystery, adds grittiness and texture, or can simply add to the overall "nighttime" look to an image - so if you want to convey any of these things then deliberately choosing a high ISO or adding grain in processing can sometimes add an extra element. (Of course the other reason is when you simply have no choice - you are stopped down as much as you can be with your aperture, and your shutter speed is as low as it can go without introducing motion blur. In that case, you simply need a high ISO to balance your exposure - but being aware of the consequences of a high ISO can mean that you can embrace the noise more fully!)
Here are some images where I've played around with noise - either by adding it in processing, or by deliberately shooting with a high ISO. It's not my usual style for sure (I wouldn't say I was normally dark, moody or gritty at all, although my husband may wish to disagree) and I'm sure it's not something that is to everyone's taste, but it really can fun to play around with photography and do something different!
Here's some where I've played around with noise:
With the image at the very top of this post, the high ISO produces softer, more muted colours, and the grain adds to the grittiness at the beach.
In this image I've stretched my camera as far as it would go, and used an ISO of 25600 so the grain is all natural. , and just helps convey that dark nightime look. (The grain doesn't show up as much in this image sized for web, but I can assure you it is very grainy!)
I added grain in processing to this image to see how it would look, since it is not a typical portrait, and I felt it lacked something. Adding noise was one of the ways I played around with it and actually I quite like the affect.
You can take an image with a high ISO to get the affect or if you just want to play around to see the affect after you have taken the image, then you can also add it in processing - here's how:
Photoshop & Photoshop Elements
Create a new layer (CMD + J on a mac, CTRL + J on a PC) then go to FILTER, then NOISE, then ADD NOISE. You can change the amount by moving the slider, UNIFORM will make grain random (looks less fake) and GAUSSIN will apply more contrast. MONOCHROMATIC changes the grain to black and white only and removes colour noise.
Go into the DEVELOP module, and scroll all the way down to the EFFECTS PANEL. You will see three sliders under GRAIN - amount size and roughness. AMOUNT will determine how much grain in there, SIZE determines how large the flecks of grain are, ROUGHNESS is the patten.
For both options, you just really need to have a play with the sliders to see how you like it! I would leave it as a last step in your processing though, as what you do after that will affect the grain (cloning or extending canvases is especially hard to do after you've added it)
Having a play with the images you take is a fun part of photography, so don't be afraid to experiment!