Let's calculate how much power you can get from the Arduino UNO's +5V pin. The UNO itself uses about 30-40 mA. Let's call it 50 mA to be safe.
On USB power the limiting factor is likely the 500 mA Polyfuse. That means you can draw about 450 mA from the +5V pin before the fuse opens. (Disconnect power briefly to reset the fuse.)
On DC power (Vin or power jack) the limiting factor is the 5V regulator on the board. The maximum current the regulator can provide is 1000 mA, but in practice the limitation is how much power the regulator can dissipate before overheating.
Maximum power dissipation (PD) is maximum junction temperature (TJ(max) =150°C) minus the ambient temperature (TA, typically 25°C) over the thermal resistance, junction to ambient (RJA). The thermal resistance is 67°C/W for the DPAK package used on the Arduino. Fortunately the big copper pad on the PCB acts as a tiny heat sink and lowers the thermal resistance to 47°C/W. This makes the power limit 2.66 Watts (125°C / 47°C/Watt).
The current through the regulator is the same as the current drawn from the regulator. The voltage drop across the regulator is the source voltage minus the output voltage (5v). The power (Watts) dissipated in the regulator is the voltage drop times the current.
To draw the full 1000 mA current from the regulator you have to keep the voltage drop low enough that the power dissipation doesn't go above 2.66 Watts:
2.66W / 1000 mA = 2.66V dropped = 7.66V source
So if you want to draw the maximum (950 mA) from the +5V pin you should use an input voltage below 7.66V (and above the 6.2V (Vin) or 6.8V (power jack) minimum). That is why 7V is an excellent choice for source voltage.
Available power goes DOWN as you increase the source voltage above 7.66V. For example, if you feed 12V to the Vin pin the voltage drop across the regulator is 7V (12V-5V). To keep the power dissipation below 2.66 Watts you can draw at most 330 mA from the +5V pin.