There are 4 basic options when it comes to staying connected overseas:
BYOP: Bring your own phone and sign up for an international/global plan with your current carrier before you leave. For most travelers, who don't visit Europe often, it's easiest to set up your own mobile phone with a basic international calling and/or data plan that's customized to your needs. This way you’ll travel with all of your contact information and your US phone number, making it easy to connect with folks back home. Don’t forget to turn off this service when you get back home.
BYOP + SIM: Bring your own phone and buy a European SIM card. This option requires that your device is unlocked (talk to your carrier). Once you get to Europe, you can buy a European SIM card to insert in your phone (or into a cheap mobile phone that you buy specifically for your trip).
SIM cards are sold in Europe at mobile-phone shops, department-store electronics counters, some newsstands, and even at vending machines. You can also “top-off” your SIM card at these locations. Before buying, ask the clerk about rates for calls within the country; to and from other countries you'll be visiting; and to the U.S. or other country you call home. Also check the rates for data use and for sending/ receiving a text message (called an "SMS" in Europe) within and outside the card's home country. Rates can vary wildly from brand to brand and store to store, and going from one country to another can be a little tricky, but for the technically astute, this is a good option.
BYOP + WiFi Only: Bring your own phone and do everything over WiFi. Travelers can avoid the cost and possible complexity of SIM cards and their carrier's international plans altogether by using WiFi. Over WiFi you can make calls inexpensively, or free, using Skype, FaceTime, Google Hangouts, etc. and you can get online to check email or browse the Internet. Not a bad option but you have to find a hot-spot first, and you have to remember to turn-off all communication options on your phone when not using WiFi.
Google Project FI: This is a fairly new service from Google that leverages multiple carrier networks to provide solid (and cheap) phone coverage here in the U.S. and in 135 other countries. It has worked well for me overseas. It only works (currently) with Google's own Android phones and it's cost-effective only for people who don't use a lot of data (data is charged by the Gb used per month). Unlimited domestic talk and text for a flat monthly fee and calls made to other countries from the U.S. are also cheap (e.g., 4¢ per minute to most European countries). Outside the U.S. you also get unlimited texting, all in-bound calls are free, and Google Hangouts to Hangouts calls are also free. Outbound calls to the U.S. from most European countries are 20¢ per minute, but using Hangouts over WiFi to call reduces that cost. Definitely worth checking out if you travel a lot overseas or have the need to call overseas from the U.S.