There are four terms (at least) that every cocktail makes will be familiar with when it comes to making cocktails - bending, mixing, mudding and shaking. These are terms that are essential to any cocktail maker, methods that define the cocktail making world. If used wrong, these methods could result in disastrous cocktails, and ruined parties.
Most cocktails can be stirred or skaken, apart from ones containing eggs or fruit. James Bond, for example, took his vodka martini shaken, even though most cocktail purists will claim this mix should be stirred (because stirring can bruise the spirit, and detract from the flavor). Each method for making a cocktail involves different techniques that result in subtle differences in how a drink tastes, and will also make a noticeable difference in how a drink looks. By all means, experiment... but know that the methods provided in most recipes are tried and true for a reason.
Blending cocktails is just like blending food, in as much as some recipes will require you to blend them together into a single consistency. Shaking won't achieve the same effects. Blending also adds texture to the cocktail by including air in the mix, leading to froth forming. Be careful not to overblend, as this will reduce your cocktail to a watery mess. Another good hint is to avoid putting cubes of ice into the blender, first crushing or breaking the ice.
Mixing is usually used for cocktails with clear ingredients. Often it involves a jug or tall glass with a lip for pouring. Ice cubes are places into the jug, and then the mixers are poured in, at which point the long stirring spoon is used to gently stir the mix until it is chilled (which is indicated by condensation on the outside of the jug or glass). Make sure to be extra gentle with fizzy mixers, as this will preserve the texture and avoid stirring it out.
At the end of most mixing spoons there is a round knob. Most people assume that it's for decoration or purchase, but this is untrue; it's called the "muddler". Old fashioned mixes - such as the Old Fashioned - take advantage of the muddler, which is used to crush certain ingredients such as the leaves of mint in a Mint Julep.