Catkins use the wind to fertilise female flowers
Catkins are flowers but since they lack any kind of petals they don't look like flowers as we popularly understand them, such as daffodils and tulips. Many catkins look soft, furry and fluffy and it is not surprising that their common names refer to cat's, kitten's and lamb's tails. 'Pussy willow' continues the cat theme.
It is a characteristic of animals that most of them are able to move about. Similarly, it is a characteristic of plants that most of them are not able to move very much; most are rooted to the ground. This poses a problem when it comes to reproducing. Animals can move about and mate with each other but plants cannot as they are stuck in the one spot.
Through the process of evolution, plants have overcome this problem in a bewildering variety of ways by using some external agent to carry sex cells from one individual to another. Catkins use the wind to carry male pollen grains to fertilise female flowers.
To maximise the impact of being wind-borne, many of the plants that are wind pollinated are tall; in other words, they are trees or tall shrubs. Hazel, birch, alder, poplar and oak all have catkins. Their height advantage is further aided by the fact that that the catkins appear early in the year before the leaf buds open. Some catkins appear in March but they are all gone by May when leaves have opened. Furthermore, many have long dangling catkins that shake and dance in the wind casting pollen on the passing breeze that blows unhindered through the bare branches.
To add even further to the chances of success many males produce very light, dust-like pollen that carries over long distances and they produce it in vast quantities as the more they produce the better the chance of some surviving. Female catkins hang in the expectation of catching a passing pollen grain blowing by in the wind.
Willows have catkins but they are evolving to be insect pollinated. They rely very little on the wind. A few herbaceous plants produce catkins too the most notable example being the Common Nettle. However, the species doesn't depend on the wind as it reproduces vegetatively very successfully using underground rhizomes and/or stolons.
One of the downsides of having all this pollen dust blowing about is that some people are allergic to it and suffer hay fever symptoms during the annual season when catkins are active.