Nginx location priority

It fires in this order.

1. = (exactly)

   location = /path

2. ^~ (forward match)

   location ^~ /path

3. ~ (regular expression case sensitive)

   location ~ /path/

4. ~* (regular expression case insensitive)

   location ~* .(jpg|png|bmp)

5. /

   location /path

There is a handy online tool for testing location priority now:
location priority testing online

From the HTTP core module docs:

1. Directives with the "=" prefix that match the query exactly. If found, searching stops.
2. All remaining directives with conventional strings. If this match used the "^~" prefix, searching stops.
3. Regular expressions, in the order they are defined in the configuration file.
4. If #3 yielded a match, that result is used. Otherwise, the match from #2 is used.

Example from the documentation:

location  = / {
  # matches the query / only.
  [ configuration A ] 
}
location  / {
  # matches any query, since all queries begin with /, but regular
  # expressions and any longer conventional blocks will be
  # matched first.
  [ configuration B ] 
}
location /documents/ {
  # matches any query beginning with /documents/ and continues searching,
  # so regular expressions will be checked. This will be matched only if
  # regular expressions don't find a match.
  [ configuration C ] 
}
location ^~ /images/ {
  # matches any query beginning with /images/ and halts searching,
  # so regular expressions will not be checked.
  [ configuration D ] 
}
location ~* \.(gif|jpg|jpeg)$ {
  # matches any request ending in gif, jpg, or jpeg. However, all
  # requests to the /images/ directory will be handled by
  # Configuration D.   
  [ configuration E ] 
}

If it's still confusing, here's a longer explanation.