Here is not the place or time to go into the laws of keeping kosher. If anybody who is not yet keeping a kosher home wishes to make their home kosher, they should contact their nearest Chabad house (or Chabad online, in cyberspace) and arrangements will be made for a rabbi to come and make your kitchen kosher and teach you all the laws and details.
Very briefly keeping kosher involves separation of milk and meat, eating only kosher animals that have been ritually slaughtered, soaked and salted according to Jewish law, and checking vegetables (and some fruits) for bugs (as we are forbidden from eating bugs). We also have to eat only kosher species of fish which means having both scales and fins.
One thing I would like to mention: chicken or beef liver are only kosher when they have been grilled over an open fire where the blood can drip down. They are not soaked or salted first. And therefore even if raw liver comes in a package with chicken, do NOT cook it or fry it. It must be grilled first to be kosher and only then can it be used in food.

Concerning kosher, one point worth mentioning is that we really are what we eat. Everything becomes absorbed into the bloodstream and into the body, into our flesh. And since our Divine soul, our nefesh Elokis, is enclothed within the animal soul, the natural soul of a Jew, and the animal soul is within the blood, therefore when we eat kosher it directly impacts both the animal soul and the G-dly soul, making it easier to perceive and connect to G-dliness, holiness. It affects our middot, our character traits, and our minds and hearts.

Non kosher food introduces impurity into the Jewish body and soul making it harder to feel or relate to G-dliness and also affecting our levels of compassion, chesed, simcha etc.
Kosher animals are those that chew their cud and that have split hooves. According to Chassidus, chewing the cud means thinking over very carefully before doing something. And split hooves is the idea that the feel do not totally touch the ground, meaning there is a separation between the earth (the matarial world) and the upper words, Heaven. Kosher meat has a good spiritual affect on us. Non kosher meat comes from animals or birds of prey, or animals with negative habit patterns.
The same with shellfish, for example: they are closed up in their shell which represents selfishness.

There are many aspects to kashrus but everything we eat and drink affects us either positively or negatively. We can well understand how the nutrition of food or eating healthy food affects our health. So too eating kosher affects the health of the neshomah, the soul, and also affects the health (spiritually and physically) of the body. Because a Jewish body is holy and cannot tolerate unholiness or impurity. That is why kosher food has such a strong affect on a Jew.

We once were in Paris and we saw a non Jew come out of a restaurant that was called Le Frog. We could not imagine what was being served in that restaurant (but most likely some type of frog!) and we saw a man come out and literally start jumping in the street, exactly like a frog! We told our children to notice how food affects a person and how much more so kosher food has a good affect on a Jew. Eating kosher refined food makes a person have refined middos, good character traits.

Also there is a story about a man who started having doubts in his faith. His father in law told this to the Alter Rebbe, the first Chabad Rebbe, and the Rebbe said that the man had mistakenly ingested some milk that was not chalav yisrael and it was affecting his faith. It is known according to kabbalah that chalav yisrael milk brings emunah, faith, and non chalav yisrael milk (even though it may be kosher but it is not milk that was watched by a Jew at the time of milking) can negatively affect one’s faith and can bring tim tum halev (which means a closing of the heart: a sort of insensitivity to holiness).

There is another story of the Rebbe Maharash who traveled to Paris, stayed in an expensive hotel, only to go sit next to a young man who was playing in a game room sipping wine. The Rebbe Maharash told him “Young man, non kosher wine stops up the heart and mind.” With those few words he brought the young man to repentance and to come back to his roots.

But from all these stories we see how what we eat truly does affect us spiritually, physically and psychologically. It is especially important to be very vigilant about the kosher standards of our children both in the home and outside the home. Many children go to other friends to play, for parties etc. and as a parent it is important to be sure that wherever our children go, the food is properly kosher, up to our standards because for a child in particular, the neshomah is very affected by what they eat. In order to have good middot (character traits) and to train our children properly, kashrut is a major factor.
So for a Jew kosher is the only way!