With no atmosphere, there are only the incoming solar radiation and outgoing earth infrared radiation to consider. Working under the principle, or assumption, of a system in balance we begin by writing
$F_{earth} = F_{solar}$
where $F$ represents the flux of energy, with units in our case of watts. (Recall that a watt is a joule (energy) per second. The expression of energy per time is the same as "power.") The Stefan-Boltzman formula relates intensity, with units watts per square meter, to the temperature of the body by
$I = \epsilon \sigma T^4$
where $\epsilon$ is the emissivity of the body ( the value is between 0 for nothing like a blackbody and 1 for an object that is truly a blackbody radiator), and $\sigma$ is the Stefan-Boltzman constant, which is $5.67\times 10^{-8} \frac{W}{m^2K^4}$.
The intensity most be multiplied by the surface area in order to give the flux. The surface area of the incoming solar radiation is that of a disk with the radius that of the earth. The surface area of the earth, a sphere, must be used for the out-going flux. Using the appropriate formulas for areas we have
$4\pi r^2 \epsilon \sigma T_{earth}^4 = \pi r^2 (1-\alpha)I_{solar}$
The $\alpha$ term in the formula represents the earth's average albedo. The equation above may be solved for $T_{earth}$ to give
$T_{earth} = (\frac{(1-\alpha)I_{solar}}{4\epsilon \sigma})^{1/4}$
Substituting appropriate values for $\alpha$ (33%) and $I_{solar}$ (1350), the resulting earth temperature is something like $-15^o$ Celsius, which is much cooler than the observed temperature of about $30^o$ Celcius. The next step is to include the influence of the earth's atmosphere.