On
$\dfrac{p}{1-p}$ translates a probability into odds. So $\log\left(\dfrac{p}{1-p}\right)$ is the logarithm of the odds, also known as the logit function. It has the advantage that multiplication of odds corresponds to addition of log-odds, and odds ratios to the subtraction of log-odds, allowing linear methods like linear regression to be applied
The inverse function of taking log-odds is $\dfrac{e^{x}}{e^{x}+1}=\dfrac{1}{1+e^{-x}}$, called the logistic function. This can be given a scale and location generalisation of the form $\dfrac{1}{1+e^{-(\beta_0+\beta_1 x)}}$
So with some binomial outcomes where you think the probability parameter is related to an explanatory variable, you could use maximum likelihood methods and a logistic model. The log-likelihood function you are trying to maximise then has a relatively simple translation to log-odds, and the logistic function then allows the results of this modelling to be translated back to probabilities
Let $X \in \mathbb R^d, Y \in \{ 0, 1 \}$ be a feature vector and label describing a randomly selected member of a population. Both $X$ and $Y$ are random variables. Our goal is to predict the value of $Y$, given the value of $X$. Logistic regression makes a modeling assumption that $$ \tag{1}P(Y = 1 \mid X = x) = \sigma(\beta_0 + \beta_1^T x), $$ where $\sigma(u) = e^u/(1 + e^u)$ is the logistic function (which you can think of as converting a real number into a probability) and $\beta_0 \in \mathbb R, \beta_1 \in \mathbb R^d$ are parameters of the model. These parameters can be estimated using maximum likelihood estimation.
This version of logistic regression, where there are only two possible classes (two possible values of the label $Y$) is sometimes called "binary logistic regression" or "binomial logistic regression", though I don't see why the term "binomial" would be used here.
Comment: If you want to predict the probability that an item described by a feature vector $x$ has label $Y = 1$, the model (1) is about the simplest thing you could possibly come up with. You might at first try making a modeling assumption that $P(Y = 1 \mid X = x) = \beta_0 + \beta_1^T x$, but that model is flawed because a probability must be between $0$ and $1$, whereas $\beta_0 + \beta_1^T x$ could be any real number. The logistic function $\sigma$ is the simplest device for converting a real number into a probability. (One elegant property of $\sigma$ is that $\sigma' = \sigma - \sigma^2$, which simplifies some calculations.)
If the logistic function seems like a weird or arbitrary thing to inject into our model, and if we view "odds" as a natural quantity to work with, we could express the model (1) equivalently as
$$ \log\left( \frac{P(Y = 1 \mid X = x)}{P(Y = 0 \mid X = x)} \right)= \beta_0 + \beta_1^T x. $$ But to me the equation (1) seems simpler and more clear.