Vacation apartments are often owned by people who like visiting a specific location several times or months out of a year, and who would like to earn money off their "second home" home when they are not there. They might also be single apartments in large complexes which the building owner keeps vacant (except of course for basic furnishings) and available for short-term vacationers while the other units have full-time residents.
Relax! You've made it to your spacious apartment after a long flight...
In either situation, the owners of the rental apartment typically contract with a management company or rental agency to maintain the property in their absence, arrange for the pick up and return of keys by visitors, and perhaps also to handle online advertisement, booking arrangements, and payment of rental fees.
Using a search engine like Google with such search phrases as "vacation apartment rentals Rome" (or whichever destination you will be visiting) will help you find listings in various destination cities or countries. Generally such listings will feature multiple photographs of the specific unit available, the amenities included, an apartment's precise location, its availability calendar for the coming months and of course the cost per night or per week.
Some websites will also helpfully feature reviews provided by former guests of the apartment, so you can read about the overall condition of the unit, any maintenance or other problems encountered, and other factors to help you make a decision on which apartment might be best for you.
If browsing through a site featuring many different apartments for rent - which I advise, as these tend to be the easiest sites to contact with questions, for booking, and for safe payment - I would recommend making a short list of 2 to 5 choices for your destination city in order of preference. Your first choice might not always be available, even if the online listing says it is. (Pinterest is a great way to keep track of your different options!)
Enjoy a room with a quiet view in Venice...rent an apartment and skip the hotels!
If your preferred unit is confirmed to be available and a quote is given on the cost for your total stay, you will generally have to make a deposit payment to secure it, typically 20-50% of the overall anticipated bill. This deposit may or may not be refundable, partially or totally, so be certain of your travel plans before proceeding to this step. You may then have to either pay the balance due at a set date before you arrive or immediately upon your arrival, and a security deposit will likely be requested as well, which will be returned to you at the end of the trip if no damage has been done to the apartment.
Note that in some cases you may have to pay the balance due to the rental agent or property manager in cash and in the local currency when you arrive, so make sure to bring sufficient funds with you. Otherwise, chose a booking agency which takes credit cards for payment in full before your trip so you don't have to worry about carrying large sums of cash with you.
You will generally need to inform the agency or apartment owner of your specific arrival time as, unlike a hotel, there will not be a 24-hour door man or desk agent on the premises to let you in. This is critical to remember, and while in transit be sure to inform your booking company or on-site representative of any delays in your travel plans if and as they happen.
Some agencies will offer to arrange taxi or other pick-up arrangements for you from the airport or train station, otherwise ask them how long on average it will take you to travel from there to the apartment.
When you arrive at the address, the apartment's agent or manager should take you through the unit to point out the different features, amenities, security systems, and other matters of which you should be aware. These details can be very important to remember, as I have stayed in some apartments with old or very tricky door locks - or where it took me three days to figure out how to turn up the heat or turn on the hot water at all! Be sure to ask how and where your keys are to be returned, especially if you will be departing after your vacation at especially early or late hours when no one might be able to meet you.