What does a modem do?

When I use my internet, it turns on and off like twice everyday, so what I do to connect back to internet is that I unplug and replug this box that i have upstairs that seems to be the main source of internet. I think its called a modem. On the box, it says Linksys Wireless Broadband Router ( it is really old, like about 6-10 years old). But the problem is that i dont want it to disconnect from internet all the time. So, If i buy another box to replace the box that i have right now, then will my internet connection be better and not turn on and off all the time? If it does, then I wanna know what type of modem should i buy? So my main questions are " Will buying a new modem make my internet slightly faster and fix it from turning on and off all the time?" and "what type or brand of modem should i buy?"