Building on @jkrideau's response a little bit, to get the plot in a single command using base R (plot and subset):
plot(x = subset(Allsites_data$Year, Allsites_data$Site == "Brownstone"),
y = subset(Allsites_data$Nitrate, Allsites_data$Site == "Brownstone"))
The kicker is this may not give you quite what you want depending on the data type of
Year in the data frame (which is one of the reasons a reproducible example is nice to have). If
Year is numeric, then you may see things like "2011.5" in the x-axis of a dot plot.
Year is a factor, you may get a y-axis that is more what you expected, and you will likely get a boxplot instead of a dot plot.
Depending on what you're looking for, specifically, you may need to do a little bit of light work on the data frame or do some additional formatting work within
plot() (or, potentially, consider switching over and using
ggplot(), but my sense is that you're not looking to dive into that world).