The first type, the K-1 visa is intended for a fiancA(C)(e) of a United States Citizen who is currently abroad and wishes to enter the U.S.A. to get married.
The second type, the K-3 visa, is intended for the foreign national spouses of a United States Citizen who wishes to enter the U.S. to wait the processing of their green card petitions while in the U.S. with their United States Citizen Spouse.
K-1 visa for fiance(e)
To be classified as a fiancA(C)(e) K-1 applicant, a foreign national must be the beneficiary of an approved visa petition wishing to enter the U.S. and marry within 90 days of entrance. Accordingly, if your fiance(e) is looking to enter the U.S. to get married a petition must be prepared and filed locally at a USCIS service center requesting the K-1 status for your fiance(e).
Some requirements include the petitioning U.S. Citizen having to establish that the petitioner and K-1 beneficiary have met in person within the two years immediately preceding the filing of the petition. The petitioner may be exempt from this requirement only if it is established that compliance would result in extreme hardship to the petitioner or that compliance would violate strict and long-established customs of the K-1 beneficiary's foreign culture or social practice.
K-3 visa for spouse
To be classified as a K-3 spouse, the alien spouse must be the beneficiary of an immigrant visa petition filed by a U.S. citizen and the beneficiary of an approved petition for a K-3 nonimmigrant visa.
This visa is intended for foreign nationals already married to a U.S. Citizen, and currently awaiting the processing of their immediate relative visa petition. Foreign nationals entering the United States as a K-3 shall be admitted for a period of 2 years.
Accordingly, the immediate relative green card process must already be in process at the time that the K-3 petition is prepared and submitted to USCIS.
Children of K visa holders
Under the K-1 Visa, a fiancA(C) of a United States Citizen is allowed to enter the U.S. with their minor children. Of course additional evidence is required for this and should be discussed with an immigration attorney. The visa that is received by the minor child of the K-1 recipient is a K-2 Visa.
Under the K-3 Visa, the unmarried children may also enter the U.S. by showing the relation to the K-3 applicant. The visa that is received by the children of the K-3 holder is a K-4 Visa.
Note that the information provided above is merely background information to give an idea of the applicability of the K visa categories. In order to prepare such petitions it is recommended to get in contact with an immigration attorney and further education yourself by visiting the USCIS website at http://www.uscis.gov/.
Note that because immigration law is federal law an immigration attorney in any state of the U.S. is licensed to prepare your immigration petitions.