Shiva is arguably of the most important deity in Hinduism. He is part of the hindu trinity and also called Shiva, Shankar, Mahesh, Shiv, natraj, natarajan and by many other names.
Shiva represents a true duality in many forms - Shiva is both a benefector (as bholenath) and destroyer (as Rudra), Shiva is ascetic but He also gives importance to Grahastha-ashram, Shiva as Nataraj relates to both death and life - so the duality hidden in various aspects of Shiva again shows the basic idea of duality in various Hindu philosophies.
Shiva is many times associated with the "destruction" aspect of nature but I must clarify that "destruction" is not a synonym for "death" in this case. Instead, destruction is more of a symbol of beginning of new things (end of things is implicitly beginning of something new and remember that every beginning also has an end and an end is not necessarily an evil thing). In symbolic terms, Shiva's Tandav form of dance is a synonym of doomsday and it is said that whenever Shiva assumes his Nataraj (or Natraj or Natarajan) form of being, it results in doomsday.
Shiva as the most powerful God in Hinduism - Shiva is perhaps the most powerful and most feared Gods in Hinduism. Again I should mention that it doesn't mean that if all Gods fight with each other then Shiva would come out as first; that is not the correct way of understanding it. Instead, Shiva represents that "aspect" of the all pervading Braham and Shiva can be seen as the sum total of all the nature's raw power in the universe.
Shiva as the ultimate Yogi - Yes, Shiva is also the most revered Yogi who controls all aspects of Yoga. I should clarify that this Yoga is not the stretching exercises that have become popular in last few decades, rather Yoga goes far beyond. Stretching exercises are a start but the goal of a real Yogi is to gain true control of all aspects of reality (body, mind, soul and beyond). Shiva masters all aspects of Yoga and thereby He masters the reality itself.
Shiva as bholenath - Another interesting aspect of Shiva is his role of Bholenath! Bholenath literally means a god who is simple minded. This is exactly opposite of Shiva as the most powerful deity to be feared. But inspite of all the power and rage, all Shiva "asks for" is true devotion and if the devotee has a pure heart then even the most insignificant offerings will please Shiva. At the same time, Shiva is very easy to displease as well (as I mention in the next section) and both gods and demons alike have suffered through his anger at one point in the mythology or other! The most notable instance being when the demon Ravana, blinded by his powers, angered Shiva and Shiva but after realizing that he is in no way comparable to Shiva (remember that Ravana had conquered death herself), Ravana came up with Shiva Tandav Strota (Strotam) that is still an unmatched masterpiece in Sanskrit literature.
Shiva as Rudra (and Rudravtars) - Rudra means "angry" and as Shiva is Bholenath, Shiva also represents the primodeal rage. So, the Rudra aspect of Shiva can be seen as the exact opposite of Bholenath aspect of Shiva. Rudra can be seen as the pure fierceness, rage and destrutive nature of the universe and probably that's what makes Shiva so fearsome too. Apart from doomsday and Tandav ntratya though, some aspects of the Rudra have incarnated periodically. There have been ten rudra avatars but the most notable Rudra avatar is Hanuman. It can be observed as Shiva's Rudra incarnation and Vishnu's incarnations represent the same duality as observed in other Hindu philosophies.
Shiva as Nataraj (Lasya and Tandav) - Nataraj, as I mentioned earlier, is another dual aspect of Shiva. Nataraj literally means the God who dances but Nataraj means much more than that! Basically the type of Shiva's "dances" can be divided into two forms - Lasya and Tandav. Lasya is related to the beginning of the universe while Tandav relates to the end of the universe. Most people know of only Tandav but Lasya is just as important in understanding the true nature of Nataraj. Again, I should note that Shiva represents the duality of nature in various ways and the Nataraj aspect of Shiva is one way of understanding this Hindu philosophy.
To learn about the philosophy behind Hindu Trinity, click -
Real meaning of Hindu Trinity
To read about the Rudra Avatar Hanuman, click -
Hanuman the Rudravtar and Bajrang Bali