We know that an inductor resists change in current and does so by creating a potential in opposition to current change; the quantity known as inductance is the ratio between voltage across the inductor's terminals and rate of change of current through it.
So, beginning with a steady state open circuit (zero current flow), when your circuit is closed there will be a voltage applied by the battery across the series combination of inductor and resistor. Since the inductor will oppose any change in current flow, the battery voltage will appear across the inductor, the voltage across the resistor will be zero and current through the circuit will remain zero... but only for an instant.
Current will begin to flow through the circuit, rising as a function of voltage and inductance; since it is flowing through the resistor, the voltage across the resistor will increase in proportion to the current through it. This will reduce the voltage across the inductor, which will slow the rate at which current will increase through the circuit, slowing the rate at which the voltage across the resistor increases and the voltage across the inductor decreases. This process continues along an exponential curve, asymptotically approaching a steady state where the voltage across the inductor has fallen to zero, all of the battery voltage appears across the resistor, and circuit current is battery voltage divided by resistance.
To summarize a few things:
- Circuit current rises exponentially from zero, asymptotically approaching a constant value defined by battery voltage and resistance
- Voltage across the resistor remains in direct proportion to current through it, so rises exponentially from zero, asymptotically approaching battery voltage
- Voltage across the inductor starts at battery voltage and falls exponentially, asymptotically approaching zero
The above is true for an ideal circuit (inductor has zero resistance). A real-world inductor will have internal resistance, and would be modeled for the purposes of analysis as an ideal inductor in series with an ideal resistor.