What does it mean to be an Ascetic?

In Christianity an ascetic is one who typically isolates himself from the world and has very little engagement with others. Such as, monks who live in monasteries who devote themselves to religious devotion abstain from the material world. They typically sell all their belongings before entering the monastery and live a simple but many times a harsh life.

In Islam one should strive to be an ascetic but its meaning and practice are much different. For a muslim, as we live our life in the material world, we should use the material world only as a means or path to heaven, to do good, promote justice, and forbid evil. Think of yourself as a ship and the water is the dunya (world). You traverse the water with the ship but if you let it seep into the ship, it slows you down. If enough gets in you eventually sink. Likewise if you become engrossed and in love with the dunya, you get distracted from your religion and your preparation for the day of judgement.
Being an ascetic doesn't mean one cannot be rich but your attachment to money and material things is in your hand not in your heart. In other words, you can relinquish it quickly without any thought. Your trust in God exceeds your trust in material things. But if it is your heart, you become attached and addicted to it. Loss of material things then causes great distress. This is the state of many muslims today, unfortunately.

A hadith Nawawi supports this:
On the authority of Abu Al-Abbas Sahl ibn Said Al Saidi may Allah be pleased with him said:

A man came to the Prophe may peace and blessings be upon him and said: O Messenger of Allah, direct to an act which , if I do it, Allah will love me and people love me.
He said: Renounce the world and Allah will love you, and renounce what people possess and people will love you.