Horseradish is one of the things often used for Maror, the "bitter herbs" on the Seder Plate for the Seder Meal on the first night (First two nights outside of Israel) of Pesach in Jewish tradition.
Although the centre of the Seder is the meal, there are various prayers and rituals around the meal, to remind the participants of the exodus of the Jews from Egypt as described in the book of Exodus.
One part of this is remembering the bitterness of the slavery of the Jews in Egypt. This is symbolised by all present eating something with a bitter and unpleasant taste. There are various customs as to what is used, which are often based around what food would be available that has the appropriate taste. Many Jews who decend from Eastern Europe - Ashkenazi Jews - use horseradish, either grated or chopped into large chunks. Romaine lettuce is also common.
The Horseradish is dipped in or mixed with Charoset - a sweet mixture that resembles cement (to remind us of the building the Jews did in Egypt). This symbolises that at first the Jews worked happily in Egypt, until it became slavery. Because of this, many people shake most of the Charoset off before eating the Horseradish.