I will not go into all details of the laws of kosher.  But 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.
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.

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!

Below is a brief summary of some of the laws and details of kosher:

All the laws of kashrut ,( kosher food)  come from the Torah. We keep these laws because Hashem commanded us to, not because we understand why. Nevertheless, in Jewish mysticism we do learn  some of the spiritual benefits of keeping kosher.

Kosher meat must be from a kosher, non predatory animal that chews its cud and has split hooves such as a cow, goat, deer etc. If an animal is missing one of those signs it is not kosher. It has to have both signs to be a kosher animal.  But just eating meat from a kosher animal is not enough to make the meat fit for Jewish consumption.

The animal must be ritually slaughtered according to the Jewish law by a knowledgable kosher slaughterer (called a shochet).  Shechita (ritual slaughter) is the most humane way of slaughtering.   If a kosher animal is not slaughtered according to Jewish law, it is not kosher for consumption.  But even having meat ritually slaughtered is not enough to render the animal ready to be eaten by a Jew.

After slaughtering the animal, the meat must then be soaked and salted properly to remove all blood. We are forbidden to eat blood of meat or fowl. (it is interesting to note that removing the blood also removes toxins and impurities, rendering the meat much healthier and cleaner). Blood is symbolic for passion and lust, as well as anger and aggressive emotions. Removing blood removes excessive lust and helps us to have purer thoughts and better control over our emotions.

Certain veins and arteries need to be removed before consuming the meat as well. This is called nikur and is done by a kosher butcher or by the shochet (the one who does the slaughtering).

The signs of kosher animals are enumerated in the Torah and the prohibition to eat blood is mentioned specifically as well.

Kosher fowl are only certain ones mentioned in Torah such as chicken, quail etc.  Turkey is also kosher. Duck has to be only certain types that are known to be kosher according to certain traditions.

Chicken and other kosher birds must also be ritually slaughtered, soaked and salted before being eaten.

Liver from kosher cows or chickens has a lot of blood and therefore regular soaking and salting is not enough. It much be broiled over an open flame to remove all the blood before it can be eaten.  They are not soaked or salted first. in fact, they cannot be put into kosher utensils until they have been kashered.  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.Only after it has been broiled can it then be cooked or fried etc.  Raw liver cannot be cooked in a fry pan or oven without first being broiled and koshered. Otherwise the pan, oven etc. would become non kosher.

Fish Must have both scales and fins to be kosher.   Having only fins but no scales means a fish is not kosher.  Shellfish are not kosher, for example, and we know that they hide inside their shell, which symbolizes being too much into themselves, selfish and self centered.

Fish is pareve (not dairy or meat). It is a custom, for health reasons, not to cook or eat fish together with meat and therefore fish is usually eaten first (like on Shabbat we eat gefilte fish and then after that we take a drink in between and then have the meat meal).

There is a prohibition in Torah not to eat meat and milk products together or to cook them together.  After eating meat or chicken we must wait 6 hours before we can eat dairy.

After eating most dairy we wait one hour before eating meat (some people have the custom just to rinse out their mouth from the dairy and then they eat meat without waiting further ).  But if the cheese is a real hard cheese (very aged like parmesan for over 6 months) then we wait 6 hours before eating meat.

Why do we separate meat and milk according to the mystical aspect of Torah ?  Milk comes from the spiritual level of chesed, kindness. Meat comes from the spiritual level of gevurah, strictness. They do not mix properly and can cause a conflict spiritually within the person if consumed together.

There is a prohibition to eat bugs or insects. That is why green vegs and other vegetables (or even certain fruits)  are washed well and  checked for bugs before being consumed.

Dairy products must come only from a kosher animal (cow, goat) and cheese must not have any byproducts of meat like rennet used in the production of the cheese.

When we buy any product such as bread or cake or cookies, we need to make sure that it is kosher because they often use oils which can be derived from animals in their products.  Everything needs to have a hechsher (a kosher symbol) to be sure it is properly kosher.

Why do kosher animals have to chew their cud?  What is chewing the cud?  First they chew their food and swallow it and then they regurgitate it (it comes back up) and they re chew it.  According to the kabbalah chewing the cud represents thinking things over and over before making a decision  etc.  Eating a kosher animal affects how we also think and behave. You are what you eat.

Why must kosher animals have split hooves? This means the entire foot does not totally touch the ground and it represents not being too connected to the material world .

Kosher food affects health of body and soul.  The Almighty created us and knows what a Jewish body and soul need to function at their optimal in terms of health , both physical and spiritual.  We are what we eat. Food becomes our flesh and blood.

If we eat a predatory animal it creates angry, aggressive attributes.  All non kosher animals or birds have some sort of negative characterists.

Everything Hashem commanded is specially tailored to a Jewish soul and Jewish body.

Keeping kosher helps us to have kosher thoughts, kinder and more refined behavior and better emotions etc. It also helps us to be more spiritually connected.