"Inkan" (印鑑) in japanese refers to both the "registered physical seal" as well as the "act of sealing/signing a document" (i.e. 印鑑します). You can have as many physical-seals, or "Hanko" (判子), as you want but for signing a legal document you need to use a registered stamp for proper validity. You can register your personal Inkan at the office of your municipality (city office).
With the registration you receive a verification card ( 印鑑登録証, いんかんとうろくしょう - Inkantourokusho) that allows you verify your stamp and register a new stamp. The verification is only useful with a Personal Identification Document.
Stamp companies, such as Hankoya, produce stamps in a way that makes them recognizable in a similar fashion way that signatures are to signature experts in other countries. Which means: it is definitely possible to forge a signature but it is maybe not as easy as you might think.
You can use un-registered Inkan's for contracts (as the other party often won't verify the stamp) but doing so is definitely problematic if there ever happens to be a legal dispute and might find you in trouble if the other party tries to verify it.
Having a registered Inkan is usually only possible for people allowed to do business in japan (which includes some, but not all Visa types). Some companies might simply ask if you have an Inkan in order to verify that you are allowed to e.g. open a company in Japan.
Theft or loss of an Inkan: In case you loose it you should immediately notify your municipality. Once its registered the documents signed by the thief will become legally invalid (as if he signed it with an invalid seal).
For businesses its common to have three different seals that are supposed to be used on different occasions.
A bigger, round one - Representative Seal - used for important, official documents. (This needs to be registered, an Inkan!)
A small, round one - Bank Seal - used or banking business.
A square one - Everyday Seal - used for daily transactions.
If you happen to make a mistake in a contract, it is proper to cleanly strike-through the passage that was wrong and then stamp it in order to verify that this change is intended.
It is problematic - generally - to have only one copy of a contract: If you sign something make sure that are at least two copies and both signed the same way to make sure it's valid.
If a document has more than one page, you should get a document binder tape (契印用テープ, けいいんよう - Keiinyou) (available at amazon here or here) and both parties need to stamp the connection between the tapes to make sure that both agree on all the documents. This is an efficiency procedure. Theoretically it would be okay to just sign each page.