I am new to statistics and I am trying to understand conceptually (and hopefully visually!) what is the difference when someone states "under the null" or "under the alternative". I guess this may stem from my lack of understanding of whether or not the null and alternative hypothesis can be based on two different likelihoods.
The exact specification of the null and alternative hypotheses depend on the test, but they all share general features.
Each hypothesis is associated with a particular probability distribution. For example, suppose you have a sample population with two variables: a type that can take values A or B, and a continuous variable. In this example, you might want to ask whether two groups of people (men and women) have different mean heights.
In this example, lets assume that height always follows a normal distribution. Our null hypothesis is that the heights for both men and women follow the same distribution. This is specified as
meanwomen = meanmen
Whereas your alternative hypothesis specifies
meanwomen $\ne$ meanmen
So you can ask for the odds "under the null" -- as in, if the null hypothesis were true, what would be the odds that you obtain your observed data? You can also ask for the odds "under the alternative" -- as in, if the alternative hypothesis were true, what would be the odds that you obtain your observed data? These would be your two likelihoods.
So the null and alternative hypotheses are not based on different likelihoods, but they each imply different likelihoods because they assume different distributions.
Since we here assumed the data followed a normal distribution, these distributions are normal. In the null hypothesis, our data all follows one normal distribution with mean equal to the observed total sample mean. In the alternative hypothesis, each group follows a normal distribution with mean equal to their respective observed group means.
As an added point, it's worth noting there are two distributions at play here: the distribution of the data, and the distribution of the sample mean. I was describing the distribution of the data above. The test above is ultimately based only on the null and alternative distributions of the sample means, which are of course functions of the distributions of the data and the sample size. In this case the hypothesis actually only specifies the mean. Frequently, such as with Student's T test, the hypothesis specifies that both the mean and standard deviation are the same.