Citizens of the European Union can access the health services of other Member States. The photo shows my "European Health Insurance Card" (I was going to black out my details - but the photo was fuzzy) - It is the size of a credit card.
The NHS website describes the card -
The European Health Insurance Card (EHIC) allows you to access state-provided healthcare in all European Economic Area (EEA) countries and Switzerland at a reduced cost or sometimes free of charge.
Everyone who is resident in the UK should have one and carry it with them when travelling abroad. Remember to check your EHIC is still valid before you travel. Applying for the card is free and it's valid for up to five years.
Presenting the EHIC entitles you to treatment that may become necessary during your trip, but doesn't allow you to go abroad specifically to receive medical care. However, maternity care, renal dialysis and managing the symptoms of pre-existing or chronic conditions that arise while abroad are all covered by the EHIC.
Your EHIC will allow you access to the same state-provided healthcare as a resident of the country you are visiting. However, many countries expect the patient to pay towards their treatment, and even with an EHIC, you might be expected to do the same. You may be able to seek reimbursement for this cost when you are back in the UK if you are not able to do so in the other country.
The EHIC is NOT an alternative to travel insurance. It will not cover any private medical healthcare or the cost of things such as mountain rescue in ski resorts, repatriation to the UK or lost or stolen property.
Brits can apply for their card here