What is Sutak?
Sutak is the period before a grahan that tradition treats as inauspicious. It begins 12 hours (4 prahar) before a solar eclipse and 9 hours (3 prahar) before a lunar eclipse, counted from Sparsha, and ends at Moksha. For children, the elderly, the sick and pregnant women it is usually kept for only 3 hours (1 prahar). Sutak applies only where the eclipse is visible: if it is not visible from your city, no Sutak is kept there.
What people traditionally do
During Sutak and the grahan many families avoid cooking and eating, keep tulsi leaves or kusha grass in stored food and water, and spend the time in prayer, mantra jap or quiet. Temples usually close their doors. After Moksha people bathe, clean the home and the puja place, cook fresh food, and give daan. Customs differ between families and regions, so follow your own tradition.
Watching a solar eclipse safely
Never look at the Sun directly, not even when most of it is covered, and not through sunglasses, X-ray film, a phone camera, binoculars or a telescope. Use certified eclipse glasses (ISO 12312-2), or watch the Sun's image projected through a pinhole onto paper. A lunar eclipse is completely safe to watch with the naked eye.
Penumbral eclipses
In a penumbral lunar eclipse the Moon passes only through the Earth's faint outer shadow and dims a little, without any part going dark. It is listed here for completeness, but it is not treated as a grahan, and no Sutak is kept.
How the times are worked out
Eclipse times come from Astronomy Engine, an open-source astronomy library, and match published eclipse timings to within about a minute. For a lunar eclipse, Sparsha and Moksha are the Moon's entry into and exit from the Earth's dark shadow (umbra), which is when the grahan is seen; the line "Seen here" shows the part when the Moon is above your horizon. For a solar eclipse, the times are for your place. Everything is calculated in your browser; your location is never sent anywhere.