A walkie-talkie is a generic term for a handheld radio that you can walk with and talk with. Often old walkie-talkie were tuned somewhere around 49MHz, sometimes on Ch14 in the 27MHz CB band and other sometimes random frequencies. Kids, families and walking groups often used walkie-talkies to keep in contact over short distances when out and about. They came in pairs, like your ones when you were a kid.

There are modern walkie-talkies pairs too, that use license free bands. PMR446 radios are sold in pairs and are the modern equivalent. You and your fishing or hunting buddies can keep in touch over reasonable distances with them. No license is needed.

Radio amateurs do use handheld radios on VHF and UHF, though they talk between each other and only need one handheld and not a pair of them. These are not walkie-talkies as such and are simply called "handhelds" or sometimes, "handy talkies". They would be similar (in concept at least) to police handheld radios. They are single units that communicate with other single units directly or via a repeater.

So, for walkie-talkies, think kids, PMR446, comms between hunters, at car rallies and used by hill walking groups and others.

"Handhelds" as used by radio amateurs are singly sold, more advanced radios. Police radios would be similar in concept, though encrypted in many countries for privacy and security sake.

As for mobile phones, they are also portable radios. Very early mobile phones were basically just portable radios that connect to a base (fixed) station that had a landline telephone service connected. These early mobile phones were just rather ordinary walkie-talkies that operated "half-duplex". "Duplex" meaning that they could transmit and receive, but "half", meaning that they could only do one of those at a time. So the user could talk, or they could listen, but had to switch back and forth using a push to talk button or something similar. 

Later mobile phones improved and became "full-duplex", meaning that they could transmit and receive at the same time. This allowed the user to talk and listen without switching with a push to talk button. Although humans can't talk and listen at the same time very well, it's an important aspect of communication as when we are talking we typically listen for sounds indicating the other person acknowledging what we are saying, or the other person needing to interrupt us.

Compared to most ham radios, even today's latest models, the complexity of mobile phones is probably much higher in regard to the number of simultaneous radio frequencies and modes in simultaneous use.

Sure, ham radios are far more complex in other ways, such as signal manipulation and bandwidth control and other technical adjustments done by the user. But the features and modes of operation of a modern mobile phone is impressive, especially given the physical size and weight of the devices.

So you can get radios in 3 different categories: receivers, transmitters, and transceivers. Walkie-talkies are a transceiver set up to operate in a particular radio service. In general, transceivers are described as base (as in a fixed location), mobile (i.e. installed in a car, but can also be used at a fixed location), or portable (these would be the HTs, or sometimes a mobile radio setup as a "manpack" in a backpack with a battery and antenna).

There are radio services that operate in different frequency ranges, for example Broadcast AM, Broadcast FM, Land Mobile Radio, Public Safety Radio, Amateur Radio, Family Radio Service, General Mobile Radio Service, Multi-User Radio Service, Cellular phone service, Maritime Radio, Citizens Band Radio, etc. Each of those services has separate rules for equipment, power levels, frequencies, and use. Depending on the service, some or all of the frequencies may be channelized.

In short, walkie-talkie is still a fairly common term used by people who aren't, but actually refers to some older military radio types. Originally, "walkie-talkies" were the backpack radios used during WWII; "handie-talkies" were the smaller (yet still comparatively large) handheld radios.