Family of distributions
Distributions that can be described by the same set of parameters make up a family of distributions. For example the normal/Gaussian family includes all of the possible distributions that can be described by the two parameters $$\mu$$ and $$\sigma$$ alone.
In brms you set the family of the probability distribution of the outcome variable with the argument family. For example, family = gaussian(), family = bernoulli().
In practice, you can think of the family as the prior probability distribution of the outcome variable.
Likelihood
This is the probability distribution of the outcome variable conditional on the prior(s).
Notation: $$p(d|\theta)$$.
Outcome/response/dependent variables
These are the variables that appear on the left-hand side of a model formula. For example, f0 in f0 ~ attitude; F1 and F2 in c(F1, F2) ~ stress.
Parameter
A parameter in a statistical model (for example intercept/mean, standard deviation, slope/$$\beta$$, etc).
A parameter used to describe a probability distribution (for example $$\mu$$ and $$\sigma$$ for normal/Gaussian distributions).
Predictors, independent variables
These are the variable that appear on the right-hand side of a model formula. For example, novel_word in reaction_t ~ novel_word; s(longitude, latitude) in temperature ~ s(longitude, latitude).
Prior probability distribution or simply prior
This is the probability distribution of the values a parameter can take, based on prior knowledge/belief, domain expertise, previous research, pilot data.
Notation: $$p(\theta)$$.