Correct me if I'm wrong, but Jews do not currently sacrifice animals like ancestors long ago. Why do Jews not currently sacrifice animals? Will there be a time when sacrificing returns once again? How are sins forgiven if an animal is not sacrificed to cover for those sins?
Moses makes it clear that when the Jews enter the Promised Land, eventually one special place shall be chosen, and that will be the only place that animal sacrifices are authorized. That place is the Temple in Jerusalem.
Today the Temple does not stand; it's unclear exactly where on the Temple Mount area the altar stood; there are numerous uncertainties regarding other prerequisites for sacrifice (such as the required garb for the priests); and let's not even get into today's hot-button political reality in Jerusalem! For these reasons and others, animal sacrifice is not done today. However on Sabbaths and holidays, our prayers clearly say that we hope to be brought back to a rebuilt Temple and offer the required sacrifices once again.
Now as the Prophets made clear time and time again (I Samuel 15:22, Isaiah 1:10-18, and so many others) -- G-d is interested in sacrifices only so much as they mean following His will. If the people are out there doing all sorts of other horrible things but bringing nice sacrifices, then sacrifices shmacrifices. Here's Isaiah (with slight paraphrasing) just so you get the sense:
Listen to the word of G-d, you people who have become as corrupt and selfish as those of Sodom: What do I need with all your sacrifices, says G-d; I've had enough with burned rams and the fat of well-fed animals; the blood of cows, sheep, and goats? I don't want it. When you approach My Presence in the Temple, who asked you to do this, trampling all over my yard?! Cease and desist, bringing false gifts; incense has become disgusting to Me. New-months, Sabbaths, designated holidays? I can't deal with the sin at these gatherings. I hate your new-months and holidays, they've become a burden I'm tired of bearing. And when you raise your hands in prayer ... I'm not listening -- your hands are full of [innocent] blood. Instead -- clean yourselves off! Get rid of the wrongdoing! Learn to do good, seek justice, improve the lot of the disenfranchised; give justice to the orphan and pursue the case of the widow. Let's make a deal, says G-d; if your sins were like bright crimson, they can become white as snow.
Hence, sacrifices as part of the worship of G-d help, but Judaism goes on without them. As far as atonement, the Laws of Repentance were described by the Talmud and codified by rabbis long ago. We are obligated to recognize our sins, confess them to G-d, regret them, accept upon ourselves to do better, and make good on that acceptance. If we do that, we've done our part.
As mentioned in other answers, we don't offer sacrifices these days because once the Beit Hamikdash was established as the permanent House of G-d we are no longer allowed to offer Korbanot anywhere else. Although the Rambam holds that we do not need the Temple to be standing in order to offer Korbanot, the Korbanot still must be offerred on the Temple Mount. Other factors however, prevent us from offering Korbanot these days. (see here).
There is a group trying to restart the Pascal Sacrifice, with much opposition (see here).
Furthermore, our Sages tell us (Menachot 110A - page 408 in this document) one who learns about a sacrifice is as if he gave that sacrifice. Also, learning Torah in general is considered as giving sacrifices (or prevents the need to bring sacrifices).
Prayer has taken the place of sacrifices in Jewish practice. Hosea 14:3 reads, "Take with you words, and turn to the Lord. Say to Him, forgive all iniquity and receive us graciously, so we will offer the words of our lips instead of calves."
And see here as well.
And see this Wikipedia article for different opinions in Jewish writings about whether and which sacrifices will resume when Moshiach comes and the 3rd Temple is built.
We actually do sacrifice animals in order to eat.
In short, we do not sacrifice animals in worship because there's no place to do it.
Please note that there were many other kinds of korbanot in the temple other than sin-forgiving. Today, for forgiveness of our sins, we can do teshuva. Also in general Hashem sends us kaparot, like bumping on the coffee table, waiting for the bus, etc. But this depends on which sin. There are those that require death as kapara.