Start by softening the butter, and letting come to room temperature. When measuring the butter, cut into small cubes. I like to cut into pieces about 1cm, ⅓″ size, as a smaller surface area will soften quicker and in turn be easier to mix with. Watch my video on how to check if your butter is soft enough. See Photo 1 below. Once the cubed butter is soft enough, place in a mixing bowl, or bowl of your stand mixer, and beat until no longer in small cubes. (See Photo 1 above). Use a hand-mixer of stand-mixer to do this, (k paddle attachment if using a stand mixer). You can of course beat by hand with a wooden spoon instead. How long it will take depends how soft your butter is and how powerful your mixer is. Anything from 30 seconds to a few minutes.
Next add in about half of your icing/powdered sugar (or about 1 cup's worth). You can pre-sieve the sugar or place a sieve over the mixing bowl. Some people don't bother sieving, but if the sugar has any clumps, and is not all an even fine powder consistency, it really is better to sieve it. (Push through the sieve with the back of a spoon to be quicker and make less mess). See Photo 2 above.
After adding in the sugar, use a clean tea-towel, or stand mixer splash guard, to cover the bowl while you begin to mix together. (See Photo 3 above). Start off on the very lowest speed, (for powdered sugar and flour, this is a must). Then have a check, and increase the speed a little until the sugar is completely mixed in with the butter.
Use a spatula to scrape any of the ingredients that might be up the sides of the bowl, not getting beaten in with the rest. Scrape them down into the bottom of the bowl, and repeat periodically during the mixing process if needed. See Photo 4 below.
Now gradually add in more sugar (about ¼ cup/30g at a time), until the mixture is now a smooth thick buttercream. (See Photo 4 above, & 5 below). Depending on the recipe and ratio of butter to sugar, some will be thicker and stiffer than others. For this particular buttercream, that will pipe well-detailed rosettes from a star tipped nozzle, as well as not immediately collapse when layering cake on top, and firm up really nice on chilling, there is slightly more butter.
Beat in the vanilla extract (try making homemade vanilla extract, it really is the best), and have a taste test. If the buttercream is sweet enough, don't add any more sugar. If the buttercream looks a little too thick, you can add in a tsp of milk at a time, until happy with the consistency. See Photo5 below. (Note how much paler the mixture is now). Like a chef, a baker should always taste test what they are making!