What does the R formula y~1 mean?

Many of the operators used in model formulae (asterix, plus, caret) in R, have a model-specific meaning and this is one of them: the 'one' symbol indicates an intercept.

In other words, it is the value the dependent variable is expected to have when the independent variables are zero or have no influence. (To use the more common mathematical meaning of model terms, you wrap them in I()). Intercepts are usually assumed so it is most common to see it in the context of explicitly stating a model without an intercept.

Here are two ways of specifying the same model for a linear regression model of y on x. The first has an implicit intercept term, and the second an explicit one:

y ~ x
y ~ 1 + x

Here are ways to give a linear regression of y on x through the origin (that is, without an intercept term):

y ~ 0 + x
y ~ -1 + x
y ~ x - 1

In the specific case you mention ( y ~ 1 ), y is being predicted by no other variable so the natural prediction is the mean of y, as Paul Hiemstra stated:

> data(city)
> r <- lm(x~1, data=city)
> r

Call:
lm(formula = x ~ 1, data = city)

Coefficients:
(Intercept)  
       97.3  

> mean(city$x)
[1] 97.3

And removing the intercept with a -1 leaves you with nothing:

> r <- lm(x ~ -1, data=city)
> r

Call:
lm(formula = x ~ -1, data = city)

No coefficients

formula() is a function for extracting formula out of objects and its help file isn't the best place to read about specifying model formulae in R. I suggest you look at this explanation or Chapter 11 of An Introduction to R.

if your model were of the form y ~ x1 + x2 This (roughly speaking) represents:

 y = β0 + β1(x1) + β2(x2)

 Which is of course the same as 
 y = β0(1) + β1(x1) + β2(x2)

There is an implicit +1 in the above formula. So really, the formula above is y ~ 1 + x1 + x2

We could have a very simple formula, whereby y is not dependent on any other variable. This is the formula that you are referencing, y ~ 1 which roughly would equate to

 y = β0(1) = β0

As @Paul points out, when you solve the simple model, you get β0 = mean (y)

---

Here is an example

  # Let's make a small sample data frame
  dat <- data.frame(y= (-2):3, x=3:8)

  # Create the linear model as above
  simpleModel <- lm(y ~ 1, data=dat)

  ## COMPARE THE COEFFICIENTS OF THE MODEL TO THE MEAN(y)
  simpleModel$coef
    # (Intercept) 
    #         0.5 

  mean(dat$y)
    # [1] 0.5

In general such a formula describes the relation between dependent and independent variables in the form of a linear model. The lefthand side are the dependent variables, the right hand side the independent. The independent variables are used to calculate the trend component of the linear model, the residuals are then assumed to have some kind of distribution. When the independent are equal to one ~ 1, the trend component is a single value, e.g. the mean value of the data, i.e. the linear model only has an intercept.