A branch represents a single element such as a voltage source or a resistor. In other words, a branch represents any two-terminal element.
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Nodes, Branches, and Loops

Since the elements of an electric circuit can be interconnected in several ways, we need to understand some basic concepts of network topology. To differentiate between a circuit and a network, we may regard a network as an interconnection of elements or devices, whereas a circuit is a network providing one or more closed paths.

Electric Circuits? It’s All About Nodes, Branches, and Loops
The convention, when addressing network topology, is to use the word network rather than circuit. We do this even though the word network and circuit mean the same thing when used in this context.

In network topology, we study the properties relating to the placement of elements in the network and the geometric configuration of the network. It’s all about circuit elements such as branches, nodes, and loops.

Branches //

A branch represents a single element such as a voltage source or a resistor. In other words, a branch represents any two-terminal element.

The circuit in Figure 1 has five branches, namely, the 10V voltage source, the 2A current source, and the three resistors.

A node is usually indicated by a dot in a circuit. If a short circuit (a connecting wire) connects two nodes, the two nodes constitute a single node. The circuit in Figure 1 has three nodes a, b, and c.

Notice that the three points that form node b are connected by perfectly conducting wires and therefore constitute a single point. The same is true of the four points forming node c. We demonstrate that the circuit in Fig. 1 has only three nodes by redrawing the circuit in Fig. 2. The two circuits in Figs. 1 and 2 are identical.

However, for the sake of clarity, nodes b and c are spread out with perfect conductors as in Fig. 1.