Music from nonwestern countries has become an increasing part of western music during the 20th century
Movies made after 2001 often use Islamic and middle-eastern music to both punctuate and define the setting of the film. Sometimes this is used simpy to create an idea of cultural exchange or "cultural globalism". Sometimes it's to give a viewer a glimpse into the cultural life of non-western communities.
I would have to say that I really do not have much exposure to music from countries outside the United States. Living in the middle of the states we are kind of isolated from a lot of things.
I see Japanese anime almost daily since all my students are enamored with it. Of course, it takes me back to the days of my youth when I watched Speed Racer on TV and wondered why the mouths didn't match the words they were saying.
There is also much Spanish culture in my everyday life. I see the migrant workers at the grocery store, here Spanish spoken in public, and experience music and cultural celebrations during visits to the local college campus.
For me, I really like music coming from a World Music label called Putumayo. I am sure you have heard of it, but I stumbled across it a few years ago with a free sampler and have never managed to get enough of their music - it is great! It has a massive range of CDs from all sorts of different ethnic and regional backgrounds and I really enjoy the exposure to World Music.
For me, I come across music from other countries in two ways.
First, I study Japanese and so I watch Japanese TV shows on the internet. When I do that, I get exposed to Japanese music.
Second, I like Norteno music from Mexico (and from across the border in the US). When I listen to Pandora, one of my channels is devoted to Nortenos.
Hawaii's not another country, but it is another source of "exotic" music that I listen to. In that case, it's either Pandora or my own collection of CDs where I encounter the music.
This sounds like a specific question from an instructor, so I am sure they are meaning for you to apply your own experience with music in this case and you'll want to keep that in mind in your own response, but the idea is an interesting one.
One source of foreign music and artists for me has been National Public Radio, which is under no ratings pressure to only play that which is popular or already well known. On weekends especially they highlight music from around the world, from what we would consider "non-traditional" styles.
In some cases movie soundtracks go off the beaten path and use foreign artists that then get international exposure. I began listening to Gustavo Santaolallo from one of his songs on the movie Babel.
A student of mine introduced me to a Persian hip hop artist, Erfan, so word of mouth is another way I am exposed to new music from different cultures. Word of mouth works on the internet too, as I've just introduced you to two new artists in this post.
"In what ways have you come across music from another country in your everyday life: TV, movies, popular songs off of the radio, etc.?"
This is a great question. But first, I have two questions to ask: 1) What country are you asking this perspective from (I assume the United States of America); and 2) Let us go back into history to question What kind of music is from another country?
Some historians will say that the only native music to the USA is that of Native Americans, suggesting that all other music would be "from another country" Of course, times have changed, ideologies have been restructured, and new traditions are sedimented into historical knowledge, all of which now affect contemporary notions of what music means today. Today, jazz is "American" but the people who propitiated it were African Americans and Latinos.
So, now that we have situated your question in this framework, we can approach an answer. The main difference between traditional, popular, commercial, and classical music is how they are maneuvered through the mass media.
Commercial and classical music makers sometimes take on aspects of popular and traditional music (improvisation, call and response, rural themes, traditional instruments, etc) because it draws in a special reference, usually something that is nostalgic). They also sometimes fuse these aspects with modern notions of music (techno beats, jazzy blended notes, electric instruments) to draw attention to the modern taste.
The main media channels are, as you mention, television shows and commercials, film music, radio programs (commercial and non commercial like NPR), musical audio and visual recordings (CDs and MTV videos), etc.
On the other hand, traditional and classical music are not as fluid in these media channels of communication. This music survives mostly by performing it, over and over again, in the way that seems to be most faithful to the way the composer wrote it, in the case of classical music, or the way our great ancestors have played it for, many centuries, in the case of traditional and folk music.
In knowing that these channels of media exists, and how they influence the flow of music across countries, one can begin to explore the ways in which they have "come across music from another country" - and whether they recognize it as music from another country, or as music of their own, from their own country.
Readings: "A Sweet Lullaby for World Music" by Steven Feld in the book Globalization, edited by Arjun Appadurai and Soundscapes: Exploring Music in a Changing World by Kay Shelemay.
Good luck in your quest!