This term is a bit of a misnomer since Rule 144A applies only to resales of securities and cannot be used by an issuer.
However, this term is generally used to refer to an offering that takes two steps:
First, an issuer private placement of securities (primarily debt, for US issuers) to one or more investment banks or broker-dealers called initial purchasers under Section 4(a)(2) or Regulation D.
Second, the concurrent resales of those securities by the initial purchasers to qualified institutional buyers under Rule 144A.
In these resale transactions, the initial purchasers act in substantially the same role as the underwriters of an SEC-registered offering.