Are electric adapters needed in the hotels in Prague and Budapest?
Try this: http://www.travel-adaptor.com/europe/european-travel-adaptors.php
You could invest in a universal adapter, which works for everywhere except Italy, the UK, Denmark and maybe Switzerland. Not absolutely sure on Switzerland, but that info will be easy to find.
Assuming you will need to plug in North American items, yes.
For Czech Republic and Hungary I never needed anything other than the simple plug adapters, North American flat blade to European round pins. That, of course, depended on anything I was plugging in was dual-voltage (NA 110v/60Hz and Euro 230v/50Hz). The same adapters have worked for me in Portugal, Spain, Italy, France, Denmark, Germany, Austria, The Netherlands, and Belgium.
The UK uses a different plug, rectangular pins with wider spacing.
Check your devices. Virtually everything with a wall-wart or other DC power adapter/charger for electronics like phones, tablets, and laptops are already dual voltage. Higher power devices like hair dryers not so much.
Hair dryers and blower brushes need voltage adapters and even then it's iffy, so plan on using the hair blower provided.
First, an adapter is a strictly mechanical device that has one country's plug on one end and another's receptacle on the other end, and the contacts are wired together. It allows you to use, for instance, a US plug in a European receptacle. Adapters do not change the voltage. Some adapters have a hole for a grounding pin on a US plug; on some the hole is actually connected to the grounding means. On others it is not - there is no contact on the male end connecting it to ground; those adapters are not safe.
A power converter is an electronic device, either a transformer (heavy) or solid state, and converter one voltage to another. A transformer produces an identically shaped sine wave at a different voltage; a solid state device essentially chops the top off the sign wave of higher voltage power, lowering the voltage for resistance devices (ex: heaters or hair dryers). The alternating current wave form from a solid state voltage converter is not suitable for electronic devices.
Devices you might use come in 3 types, single voltage, dual voltage, and multi-voltage.
Single voltage devices might be designed to run on, say, 110 volts. Putting 220V into a 110V device would probably cause it to burn up. Likewise, using a 220V device on 110V would probably not damage the device, but, at best, would give you poor performance.
Dual voltage devices are designed for either of two voltages, say 110V or 220V. They have heating elements designed for 110 volts, and use a switch to put the elements in series for 220V input or in parallel for 110V input.
Multi-voltage devices, such a USB "chargers" convert any input voltage, from 100VAC to 240VAC to 5 VDC for electronics.
Now, of the adapters shown on the link provided by Wanderlust,
the budget adapter, top left, with what is sometimes called a Euro-plug, type C, will work in virtually any receptacle on the European continent, including Italy and Switzerland. However, it is NOT grounding, and should not be used with a three prong US grounding plug. It is also limited by code to devices using no more than 2½ amps, which would exclude hair dryers. Lastly, although it has one blade slot wider than the other and will accept a US "polarizing" plug, European power, with the exception of Switzerland's, is not polarized. Never use a US device with polarizing plug in Europe.
Similarly, the Triple Outlet adapter has a non-grounding, two pin plug (type C) and should not be used with US three prong grounding plugs or with polarized plugs, or for more than 2½ amp (575W) appliances.
Lastly, the adapters shown as "budget 6-pack" are proper French/German Schuko (types E & F) plugs and are grounding. They can be used with US three prong plugs and for more than 575 amps. I happen to have a set of these. Mine do not have a wider slot for polarized plugs, which I think is good.
I've been in Prague, and as I remember, their receptacles are the French style, Schuko receptacles. The budget 6-pack adapters should work in Prague.
Every USB power supply (phone charger) I have seen does not need a ground and does not have a polarized plug.
Edit: I just found this website, https://www.worldstandards.eu/electricity/plug-voltage-by-country/. It shows the type of plug used in every country. The Czech Republic does use the French style type E plug; Hungary uses the German Schuko type F plug, so the budget 6-pack adapter will work in those countries.
For Prague, also remember to bring spare fuses for anything that has a fused plug. Our plugs here are generally not fused and the regular wall sockets don't have switches. So whatever you plug in is "live". So if you plug anything into the room and something goes pop it's either the fuse in your plug (and you don't want to hunt around for an electrical repair shop) or the circuit breaker (usually by the door).