Given product has sold x times, what is the probability that a product will sell again? I'm not sure how to look up and read about this problem, and I'm hoping you might be able to point me in the right direction. In essence, this question is about whether or not I should purchase another unit this year:
Given that I have sold 3 of item x this year, 
what is the probability that I will sell a 4th?

I think this is related to a manufacturing question of this variety:
Given that the machine has produced 3 units without failure
what is the probability that it will produce a 4th unit without failure?

In a less simplified version of this question, you might take into account that sales is a time series (that unit 1 was sold at time 0, unit 2 was sold at time 2 and unit 3 was sold at time 2.5), and attempt to predict the probability that unit 4 will be sold within time 5 - but I'm not sure I have the statistical foundation to understand how to do that quite yet.
What is this type of problem called, and what should I focus on learning to be able to answer it?
 A: I'm sure there are more in-depth answers to be given for this general problem, but for the specific problem you mentioned, this sounds like it would be straightforwardly modeled with a Poisson distribution.
A Poisson distribution models independently occuring events in a fixed time period.  In your example, you had one event in the $[0,1)$ interval, no events in the $[1,2)$ interval, and two events in the $[2,3)$ interval, you can represent the events like this: $k = \{1, 0, 2\}$.  The MLE for $\lambda$, the Poisson parameter is $\frac{1}{n} \sum k_i = 1$.  
To find the probability that at least one event occurs within the interval $[3,4)$ is equivalent to finding $1-Pr(X=0)$ (one minus the probability that no events occur).  So we can use the Poisson function's probability distribution function: $Pr(X=i) = \frac{e^{-\lambda} \lambda^i}{i!}$ to calculate this.  
$Pr(X=0) \approx 0.37$.  So the probability that at least one event occurs in the interval $[3,4)$ is about $0.63$.
A: I think that sales forecasting is too big of a topic. Without being specific, you will not get a lot of useful help here.
Having said that I'd suggest starting with autoregressive time series models, such as AR(P) or ARIMA(P,1,Q). Sales are usually persistent, i.e. next month sales are similar to this month. They are also often highly seasonal, so you may need to add seasonality to your model. In time series it's often done through multiplicative seasonal time series. 
If the forecast is short, less than a year, and the firm is not in explosive growth mode, then ARMA(p,q) maybe enough.
Sales may depend on external variables, such as commodity prices or marketing spending, you can add them in ARMAX or ARIMAX framework. Be careful with potential endogeneity issues with variables like marketing spend though.
