For large classes of examples of bicategories the 1-morphisms naturally are of one of two different types:

special morphisms that may be taken to compose strictly among themselves;

more general morphisms that behave like bimodules.

The archetypical example is indeed the bicategory whose objects are algebras, and whose 1-morphisms are bimodules between these: every ordinary algebra homomorphism $f : A \to B$ induces an $A$-$B$-bimodule ${}_f B$ and this operation induces a 2-functor from the category of algebras and algebra homomorphisms into the bicategory of algebras and bimodules, which is the identity on objects.

For usefully working with bicategories of this kind, it is typically of crucial importance to remember this extra information. A framing on a bicategory is a way to encode this.

Framed bicategories are also known as fibrant double categories, because they may be characterised by the property that $(L, R) : D_0 \to D_1 \times D_1$ is a fibration (Theorem 4.1 of Shulman ‘08). They are called gregarious double categories in DPP10.