How to represent a categorical predictor rstan?

Another approach is to use an index variable, in which case the Stan program would look like

data {
  int<lower = 1> N; // observations
  int<lower = 1> J; // levels
  int<lower = 1, upper = J> x[N];
  vector[N] y;      // outcomes
}
parameters {
  vector[J] beta;
  real<lower = 0> sigma;
}
model {
  y ~ normal(beta[x], sigma); // likelihood
  // priors 
}

and you would pass the data from R to Stan like

list(N = nrow(my_dataset),
     J = nlevels(my_dataset$x),
     x = as.integer(my_dataset$x),
     y = my_dataset$y)

It is correct that Stan only inputs real or integeger variables. In this case, you want to convert a categorical predictor into dummy variables (perhaps excluding a reference category). In R, you can do something like

dummy_variables <- model.matrix(~ country, data = your_dataset)

Which will look like this

   (Intercept) countryEngland countrySouth Africa countryUSA
1            1              1                   0          0
2            1              1                   0          0
3            1              1                   0          0
4            1              0                   0          1
5            1              0                   0          1
6            1              0                   0          1
7            1              0                   1          0
8            1              0                   1          0
9            1              0                   1          0
10           1              0                   0          0
attr(,"assign")
[1] 0 1 1 1
attr(,"contrasts")
attr(,"contrasts")$country
[1] "contr.treatment"

However, that might not come out to the right number of observations if you have unmodeled missingness on some other variables. This approach can be taken a step farther by inputting the entire model formula like

X <- model.matrix(outcome ~ predictor1 + predictor2 ..., data = your_dataset)

Now, you have an entire design matrix of predictors that you can use in a .stan program with linear algebra, such as

data {
  int<lower=1> N;
  int<lower=1> K;
  matrix[N,K]  X;
  vector[N]    y;
}
parameters {
  vector[K] beta;
  real<lower=0> sigma;
}
model {
  y ~ normal(X * beta, sigma); // likelihood
  // priors
}

Utilizing a design matrix is recommended because it makes your .stan program reusable with different variations of the same model or even different datasets.