Dinner, in the UK, is a main meal. Supper is a light snack (usually!) taken late in the evening.

Supper is the easy one. Everyone knows what supper is: snack/late/before bed.

Dinner is the problem here in the UK. For most people it's probably an EVENING meal. Certainly, you go OUT TO DINNER in the EVENING, and OUT TO LUNCH in the MIDDLE OF THE DAY somewhere.

However, many people call the meal in the middle of the day - DINNER. They have their dinner about 1pm or so. Generally it's working class folk who label the meal in the middle of the day 'dinner.' They will probaly go home for their TEA (another cooked meal) about 6pm. They may have a light supper before they go to bed.

Middle class people usually have breakfast, lunch, and dinner in the evening. Depending on the time of their dinner they may have a supper too!

I had to research this for some Korean & Japanese ESL learners in college a few years ago. What most people are saying is true, the use of the word "dinner" varies by region. It can refer to lunch (midday meal) or supper (evening meal) depending on where you are. There are two reasons it's so varied: 1) It comes from the word "to dine" which just means eat and 2) historically speaking it refers to the largest meal you will eat in a day. Some people eat a larger midday meal and others a larger evening meal.

In the town I grew up in on the Canadian prairies, the use of the word dinner would vary from house to house, so if the meaning isn't clear by context most people will just say lunch or supper. Some of my American friends from California laugh at me for saying "supper" instead of dinner because they think it sounds old fashioned!

According to Dictionary.com:
Supper is a light evening meal - served in early evening if dinner is at midday or served late in the evening after an early-evening dinner. Either way, it is regarded as the last meal of the day. Dinner is the main meal of the day, served either in the evening or at midday.

Where I live in New England (USA), we tend to use dinner and supper interchangeably.  

In the US, the use of "supper" versus "dinner" is indicative of age and/or social class. You would rarely hear an educated and/or upper class person using "supper". In most areas, only less-educated individuals and possibly elders would use the term supper. People in the Deep South (southeastern region of the US) are more likely to say supper, regardless of social class, but their regional dialect differs considerably from the rest of America. 

If traveling in the US, say dinner. It will be appropriate in every situation, but supper might not be appropriate or might seem out of place in some situations. 

In the United States the terms are now largely used interchangeably. Older people, especally in the American South, call the mid-day meal dinner and the evenIng meal supper.