This mostly depends on the size of the place you’re referring to. In fact, you could also add "à" in your question.

Here are some general rules.

En: for large places with a feminine name

In general, we use "en" for large places, like countries or regions, when they are singular and feminine names or begin with a vowel. That is in fact, countries or regions for which you would use "la" or "l’" to refer to them:

Countries: En France, en Allemagne, en Italie, en Grande-Bretagne, en Iran (masculine name, but begins with a vowel)

Regions: En Auvergne, en Île-de-France, en Picardie, en Californie, en Corse

Au/aux: for large places with a masculine or plural name

For other country names, we use "au" or "aux" (for plural). That is in fact, countries or regions for which you would use "le" or "les" to refer to them:

Countries: Aux USA, au Canada, au Japon, au Bénin

Regions: Au Kamchatka, au Connemara

À: for local places

For localized places, like cities, towns or villages (and sometimes small countries), we use "à":

Cities: À Paris, à Lyon, à Bordeaux

Small countries: À Monaco, à Taïwan, à Cuba

Note that Andorra is not bigger than Cuba, but we say "En Andorre".

Dans: for very localized places

For very localized places, like a house or a box, we use "dans":

Dans ma maison, dans cette boîte, dans le jardin

Exceptions to the rules

There are however exceptions (otherwise it wouldn’t be fun):

Regions: Dans le Berry, dans le Limousin

Cities: En Avignon

Generic places: Dans le ciel, dans la mer

Expressing time

To express a lapse of time to wait for, we always use "dans":

Je pars au Canada dans 3 jours, dans 1 mois, dans un siècle

To express a lapse of time necessary to achieve an action, we use "en":

Le tour du monde en 80 jours

When replacing in, the situations where you have to use en instead of dans are the following:

• When depicting a situation, without articles

  My son is in sixth grade
  Mon fils est en sixième

• For some locutions

  In front of…
  En face de…

• For feminine continents, countries and regions names (all continents are feminine, and almost all countries are feminine, PLUS some masculine country name are also used with en)

  I’m going to Germany
  Je vais en allemagne

Exception: Quebec (masculine). note the use of au instead of dans used for large places like countries / regions

I’m going to Quebec
Je vais au Québec

• For season names

  I go to the pool in summer
  Je vais à la piscine en été

• And some other very special cases

  In return
  En contrepartie / En retour

Note that this list only covers the cases where en is used when translating from in. You’ll find other uses of en in other occasions.

In situations where “en” and “dans” are usually translated by “in” a useful guide is to note that “en” is used immediately before a noun, whereas “dans” is followed by an article. Compare

aller en ville

aller dans la ville

But as with all such guides beware:

en l’absence de (in the absence of)

Here are a few useful expressions that use “en” which would most likely be translated by “in”.

être en danger (to be in danger)

être en train de (to be in the process of)

croire en Dieu (to believe in God)

être en bonne santé (to be in good health)

être en difficulté (to be in difficulty)

In general “en” is used for abstract ideas while “dans” is used for physical locations.