During the year on Shabbat we eat Challahs that are braided. One reason for this is because each strand of the challah represents one of the six days of the week that we have completed, and now turn all those strands into Shabbat enjoyment, as if to say, all the six days of work are now complete, let us rejoice in the culmination of these days and enjoy the fruits of our labor by thanking and recognizing G-d on this very special day.
On Rosh Hashanah however we have a custom to make round as opposed to braided challahs. Why is this?
One opinion is that since on Rosh Hashanah we are crowning G-d as King over us, we make a challah that looks like a crown!
The word for year in Hebrew is 'shana'. This word is related to the word in Hebrew for 'repeat'. The year for us has a cyclical nature. We travel through the year until we reach the end, and start again. The challah represents this continuous circle of life, by reminding us that the Jewish year is about to be completed and we need to focus on restarting and improving the new year from it's start. The word shana, also means to 'change'. As the year comes to a close and we restart the cycle, we also want to make the upcoming year a year of change and newness.
Circles play a big part in Jewish life. We place a circular ring on the bride under the chuppah, we also dance in circles at weddings and other celebrations. King Solomon was famous for wearing a ring with the words "this too shall pass" inscribed upon it. Rabbi Avigdor Miller used to say this was because when life was challenging for him, he would look at his ring and think, "life is tough now, but it will be better soon such is the cycle or ring of life." So too when things were going well, he would look at the ring and remember that even the good times do not last, and a hard times are always around the corner, such is the cycle of life.
On Rosh Hashanah we look at the challahs and remind ourselves that no matter how bad or good the last year was, we are part of the circle of life, and the new year will bring good times and also challenges with it. We pray to G-d that he protects the upcoming year, and gives it all blessing just like the blessing we make over the challah.