For many countries in the East, there is no such tradition.  After all, signatures can be easy to counterfeit.  Instead, seals and stamps are used.  This practice originated in China, but is still practiced to this day in many asian countries.  This is called an "Inkan" for formal purposes, and sometimes referred to as a "Hanko" for informal purposes.  In English, it's usually called a "Stamp" or "Seal".  You can find more general information about the history of asian seals using the links below, but our focus here is the practical use in Japan.

It depends, many people live in Japan for years without having one.  Other people need one relatively soon.  Even in many situations where an Inkan is required, some institutions may make a special exception for foreigners.  On the other hand, the "Proper" way is usually to get an Inkan, and some places may refuse to offer your some service without one.  Even in cases where you might be able to slide by without one, people will be more comfortable (and impressed!) if you do.  It lets them know that you are just like them and know what's going on.  On the other hand, once you start using one, you may have to continue using it.  

There are two types of Inkans:

1. Unregistered Inkans

2. Officially Registered Inkans (Jitsuin)

Most Japanese people have both.

Unregistered Inkans are often used for things like signing for packages, etc.  Often many small contracts can also be "signed" without proof of registration.  (As they are likely to check ID, etc., as well.  In that case, the Inkan merely proves your seriousness, not identity.)  In many of these cases, you can get away with a signature, especially if you are a foreigner and/or don't have an Inkan.  

On the other hand, serious contracts for large amounts, especially things like mortgages, almost always require an officially registered Inkan.   Usually in these cases, there is no getting around having an Inkan either.  Opening Bank accounts at some banks requires a registered Inkan, while it is optional at others.

It is generally considered that a registered Inkan confers authority to enter into contracts on your behalf, and you should guard it with your life.  What's more, it's possible to have a copy Inkan made from an impression, which means that it's safer if you use two separate Inkans for things like package signing, and more serious business.  Also, for example, if you used an Inkan to open an account, you will need it to make modifications, so don't lose it.  

The creation process is simple enough.  There are many Hanko shops all around Japan.  You simply visit one, and pick out your design, size, material, style/typeface, and what text you want.  Inkans can be made out of Metal, Wood, Jade, Ivory, or nearly any material.  Custom made Inkans range greatly in price, depending primarily on size and material.  They can be anywhere from around USD $25 or a basic one, to many hundreds of dollars for a fancy one.  Inkans are often used as status symbols, with CEOs and such carrying super expensive Inkans.  If you go to a place like Tokyu Hands to have an Inkan made, they will often have interesting novelty options, like Inkans with combination locks or projectors.

Generic Inkans can be bought at the dollar store.  You really don't want to use these for official purposes, since they are all the same, and the government won't let you register them.

Even if you have a cool idea for putting some Kanji for your Inkan, note that in order to use it for official business, you will have to have it match your legal name on file with the government.  You can create a legal alias, if you want, but you should make sure that's approved prior to spending the money on the Inkan.

Usually the shops will ask you to come back in a few hours, or the next day to receive your shiny new stamp.  

Once you have your Inkan in hand, you can get it registered.  In order to do this, you go to the local government ward office where you are registered.  You will have to pay a small fee, and they will take an impression of the Inkan for their records.  They can produce an official paper certificate, and/or a card.  The paper certificate is more useful, as it has your name and a copy of the impression of the Inkan on it, and you can show it to a bank/realtor/landlord/employer to prove that the Inkan you are using to sign a document is indeed yours.  The card, on the other hand, often allows you to get new copies of the certificate easily.  (Some places ask to keep an original copy of the certificate).

Why Register?  You don't need to register your Inkan in order to use it for things like signing for packages, etc., but you may well want to register an Inkan you use for important transactions.  This allows you to verify that it is indeed your inkan by having city hall issue an Inkan Certificate.  

Note that many people have multiple Inkans:

1. One for general (unimportant) use, such as packages, etc.  This may be a generic store bought pre-engraved one.

2. One for banking (Registered with the bank)

3. One for other important transactions (aka "Jitsu-in"), often registered with city hall.

Interesting fact:

There are Inkan vending machines, were you fill in the information on the screen, and it will engrave an Inkan automatically on the spot.  In order to preserve uniqueness, you can select the font itself, the size, etc.