Difference between model / line? I have a dataset. I'm asked to write out the linear regression model. I'm also asked to calculate the least squares regression line. What's the difference?? [[EDIT]]
 A: What the model consists of depends, partly, on what you're doing, and partly on whatever conventions for writing a model you're using. I'll give some examples, assuming you want simple linear regression.
Typically a (statistical) model will give a distribution for the $y$ values, like so:
(i) $y_i  = \beta_0 + \beta_1 x_i + \varepsilon_i$; where $\varepsilon_i \stackrel{\text{iid}}{\sim} N(0,\sigma^2)$ ; or
(ii) $y_i \sim N(\beta_0 + \beta_1 x_i,\sigma^2)$ , independent of the other $y$'s
Sometimes the distributional assumption of normality is not made (it's only required for the normal theory hypothesis tests and intervals), in which case something like:
(i)' $y_i  = \beta_0 + \beta_1 x_i + \varepsilon_i$; where $\varepsilon_i \stackrel{\text{iid}}{\sim} (0,\sigma^2)$ ; or 
(ii)' $y_i \sim (\beta_0 + \beta_1 x_i,\sigma^2)$ , independent of the other $y$'s
where in this case the $\sim (,)$ is indicating that the observations are from some unspecified distribution with mean and variance given in the parentheses.
There are other ways the model might be written. Look to your notes or textbook, which will contain the examples you need to emulate.
Note that the model is not a line. The model has a line joining the population mean of the $y$'s at each value of $x$, but the model does not merely describe the mean. It describes the distribution of the  data, or in the simpler case, at least its mean and variance.

The least squares line will be a line of the form
$\hat y_i = b_0 + b_1 x_i$, where the $b$'s are the parameter estimates. You'll replace them with their particular numeric values when you estimate the model.
This will be a line you can draw on a plot, for example, or write down as an explicit equation for a line with numerical intercept and slope.
A: For me, the distinction is that the model would be something like
$$y_i = \beta_0 + \beta_1x_i$$
in the case of a simple linear model with one independent variable, whereas calculating the least squares regression line would involve estimating the parameters $\beta_0$ and $\beta_1$ via least squares.
