This phrase, like many other number references in Judaism, is somewhat of a misnomer. The ten days are really seven, because it includes the two days of Rosh Hashanah as well as Yom Kippur. The Shome Esray prayer, means "18," but actually consists of 19 blessings. The seven days of shiva (which means seven) are really five and a half since they include the day of the funeral and ends the morning of the seventh day. The rule that a person who misuses sanctified property is penalized by adding a fifth of the value actually results in a person paying a quarter of the value.
One of the defining features of the ten days is are the six changes to the prayer. Four of the changes involve insertions to the various the first, second, second to last and last blessings of the prayer. The other two changes involve the substitution of words in the "chatima" at the end of the third and eleventh blessings.
The former is the most significant change wherein the words "Hamelech Hakadosh" (the Holy King) is substituted for "Ha'el Hakadosh" (the Holy God). If you don't make this substitution, your prayer is considered invalid, you have not fulfilled your prayer obligation, and you must repeat it.
There is also two slight changes to the every kaddish. One extra word is added. Instead of saying "le'elah min kol berchata" (above all blessings), you say "l'elah u'lelah min kal berchata" (above and above all blessings). (Some versions leave out the vov and read it as "l'elah l'elah".) In addition, at the end of the kaddishes, you substitute "oseh hashalom bimromav" (the one who makes peace in heaven) for "oseh shalom bimromov" (the maker of peace in heaven).
Given how often you say the prayers (three times a day) and recite kaddish (at least eight times a day), it is not easy to remember to make all these changes.