Why is it important for Jews to observe shabbat?

The Shabbat is a time to rest, to stop the hectic and stressful lifestyle of the weekdays and sit back, to reconnect with family and community, and to inject a dose of Judaism into one's modern milieu.

Here's a brief picture:

There are two main aspects to Shabbat observance: what we do and what we don't do.

What we don't do: we're not permitted to work on the Shabbat (Exodus ch.20). This includes 39 categories of productive interaction with the world, such as planting, writing, kindling fire, etc. (Talmud, Shabbat 73b).

What we do: candles are lit, customarily by the lady of the household, around 20 minutes before sunset on Friday afternoon. We then attend synagogue for the Friday afternoon prayer (mincha), the kabbalat Shabbat (ushering in of the Shabbat), and the Shabbat evening prayer (maariv).

On Shabbat morning, we again attend synagogue. The services are longer than on weekdays and include prayers as well as reading the weekly Torah-portion. There's often a kiddush (refreshments) afterwards, and congregants have a chance to schmooze (to talk). Towards the late afternoon, there's another (short) service (Shabbat mincha).

After Friday night services and on Shabbat morning after services, we come home, often with guests, make kiddush (blessing over wine), and have a leisurely multi-course Shabbat meal including singing and Words of Torah. Customarily, that week's Torah-reading (parsha) will be a topic of conversation; and the children of the family will have a chance to speak of what they've learned in school.

After that, Shabbat is a quiet time: no phones, radio or TV (etc.); just schmoozing, taking walks, visiting friends, reading, learning Torah, playing Board Games, etc. Husband and wife, in particular, finally have a chance to be together after a hectic week.