In your first schematic, the nodes are mislabeled -- there is no such thing as a node current. Current only flows in loops. So you can't label a node "5V @ 1A", you can only label the node "5V".
I've redrawn your schematic in conventional form, with the energy flowing from left to right and the higher voltages towards the top.
simulate this circuit – Schematic created using CircuitLab
There are three nodes (a node is a "place that has voltage"):
Node voltage V1 is the voltage at BAT1(+) terminal. and one terminal of R1.
Node voltage V2 is the voltage at one terminal of R1 and one terminal of R2.
Node voltage V3 is the voltage at one terminal of R2 and at BAT1(-) terminal.
This is a lumped-constant model, so we're ignoring the minor effects of wiring resistance and inductance, and just assuming that the voltage is the same along the entire length of a wire.
There is one mesh loop (a mesh is a "loop that has current"):
The mesh current i flows through BAT1, R1, and R2. The same current flows through all of these components.
This is a DC circuit because the node voltages and mesh currents are steady, and do not change. For changing signals there are AC circuit analysis techniques. But for DC circuits, the three basic laws that always apply are Ohm's Law, Kirchoff's Current Law (KCL) and Kirchoff's Voltage Law (KVL).
KCL applies at each node, because the total of all currents "in" and all currents "out" of that node, must be equal (i.e. the algebraic sum of all currents going "in" must be zero). This gives one equation for each node.
KVL applies around each mesh, because the algebraic total of all voltages around the loop must equal zero. This gives one equation for each mesh.
Ohm's law applies at each resistor (and this is where the AC analysis gets more complicated). Ohm's law relates the voltage between the two nodes, to the mesh current that flows through the resistor. This gives an equation.
Put all these equations together, do some algebra, and you can determine the voltage at every node and the current through every mesh. For DC circuits analysis, that's how the circuit is "solved".