The first "city guest card" appeared in the early 1980s in Stockholm. Since then, such bonus programs have become so popular that even tiny European settlements acquired their own "plastic".
The simplest type of card is essentially a regular pass. You can ride public transport for 6, 24, or 48 hours with unlimited trips. This offer is unlikely to interest those who stay in the city center and prefer leisurely walks. Conversely, such cards are absolutely necessary for travelers who plan to crisscross the city in a day (or 3-6 days). Passes are indispensable if the city has a complex and confusing public transport system: figuring out which zone a museum is in, then determining how many zones a ticket needs to cover to get to the next—that's a waste of time. But such a "travel pass" won't eliminate all hassles.
For example, in the French capital, there is Paris Visite - a card for children and adults for various numbers of days. It allows travel on buses and the metro, but only within zones 1-3, so you can't get to Versailles, Disneyland, or Orly Airport with it. Meanwhile, besides this tourist pass, Paris has a huge number of various more favorable travel cards for residents.
The most common type of "guest card" is a one-day pass that allows free or discounted entry to several museums. This is beneficial primarily for those without discounts: student discounts in Europe are so large that buying such a card makes no sense for them. If exam times are long past, it's good to at least roughly know museum prices. Not everyone can visit several exhibitions in a day. Maybe it's worth realistically assessing your strength, just buying a ticket to a single gallery and ending the cultural program there? And it's hardly worth messing with such offers for those who prefer to sleep in, watch TV, then go to the nearest shopping mall and spend the rest of the day there.
Usually, when calculating the card price, entrepreneurs assume that a tourist will visit three museums and use transport three times per day. The Paris Museum Pass for 2, 4, or 6 days allows entry to more than 60 of the most significant museums and monuments in Paris. But who is capable of that? When buying it, keep in mind that some museums require a whole day. In France, they are very lenient towards youth under 25—showing a passport is enough to get many discounts, and girls and boys under 18 don't even need a ticket for many museums. Additionally, some museums let all visitors in for free once a week.
Some cards completely replace tickets, others act as "discount coupons". Transport passes that also promise "discounts to some museums" while clearly overpricing the fare usually only grant access to minor galleries that no tourist would ever willingly visit. Cards for three to six days may offer discounts not only on museums and transport, but also on hotel accommodation, restaurants, and cafes. They don't solve the queue problem—you'll still have to stand in line for tickets—but they may help choose a hotel and restaurant. However, there's no guarantee that the prices at these establishments will satisfy a poor student, or that the discounts will pay off. Such an offer appeals to those who know nothing about the city, don't want to spend time searching, have a decent amount of money, and trust local authorities. But why then need discounts? Not everyone wants to eat and stay in prescribed places: for whimsical, spontaneous natures who often change plans depending on mood, such a card is unlikely to be useful, and they probably won't last six days in one city anyway.
When buying a card, pay attention to when the offer starts and ends, whether it includes visits to nightclubs or everything expires at midnight? There are cards valid for an entire region or even a whole country. For example, the German Hamburg CARD can be used throughout the region, and the ABC-Berlin WelcomeCard "works" both in Berlin and Potsdam.
For children, special cheaper cards are usually sold that allow shopping: for instance, if you need children's clothing, the store will give a discount with the card. And sometimes one adult card can cover several children. Group cards are popular among travelers (and not necessarily family ones). Buying in bulk is always cheaper—young people understand this and get to know each other right at the information kiosks. For example, a one-day Hamburg CARD costs 8 euros, while a group card (maximum five people) costs 11.8: savings start from the second person.
Cards are convenient for those attending various events. Large crowds, queues, jostling and confusion—all accompany mass spectacles. By buying a card in advance, you can save not only money but also nerves. The authorities of the Austrian state of Salzburg took care that fans arriving for the Euro-2008 championship not only supported their favorite team but also got to know the country, and therefore a special Fan Card was issued. With it, you can travel unlimited throughout the region, visit museums, exhibitions, and public events—it provides entry to a total of 190 sites.
You can purchase 'guest cards', as a rule, at train stations, at special kiosks in the city, in some hotels, and of course, at tourist information offices - a very useful place for a traveler. It's better to start getting acquainted with the city from them: get up-to-date information about concerts, buy souvenirs cheaper, get free maps (in the nearby bookstore you would have to pay a couple dozen euros, or even more, for almost the same ones) and ask in detail about discounts. If you don't have time to look for tourist offices, most modern bonus cards can be bought online.