As technology has improved, some devices typically described by one term have evolved to include features described by one or more other terms, so you kind of need to dig-in to the feature-list of a new and unfamiliar device to get a more accurate sense of its capabilities.
An Audio Recorder is a device that records audio to some media. There are many examples of recorders today that are also Mixers and Audio Interfaces, some being portable, some offering multitrack recording (recording multiple tracks at the same time), some being able to overdub, etc...
An Audio Sequencer is a device that can be programmed to produce a sequence of notes/signals/data. An Audio Sampler is a device that can record/edit a "sample" of audio and map it to an actuator for later performance (for example, to a tactile pad that is hit with a stick). Today, there are modern sequencers that are also samplers, multitrack recorders, audio interfaces, etc... One software-based evolution of the sequencer is a digital audio workstation (DAW), which is what GarageBand is.
A Looper is another device worth mentioning that is somewhat of a specialized sampler/sequencer. Some loopers can move from different recorded tracks in time (sequencer-like), but the primary focus is on layering, where a sample is recorded by adding it to a previous sample in a given track (some devices buffer this data, so that a newly added sample can be removed if the user didn't like the result). The result is then played back over and over again (looped).
Here's an example to (hopefully) illustrate the variance some of these terms can have. I have an analog keyboard synthesizer that includes an arpeggiator and a sequencer. Now, yes this sequencer has some editing capability to help in the creation of a sequence of notes, but it's not as general purpose as a DAW, it cannot perform multitrack recording, it only "samples" some aspects of a note from the keyboard (like pitch and duration, but not timbre), etc... (It's an analog sequencer, it's meant to be fast, interop with compatible electronics, and in this case also improve the instrument's user experience.)
If you're looking for a standalone piece of kit that is most like a DAW, you're probably looking for a standalone "sequencer+sampler". Standalone can be relaxed if you're bringing a laptop with a DAW. If recording of a small band is more your thing, you might want to look at a digital "mixer+recorder+audio-interface". If you're just looking to put things together "on the fly" (experimenting/exploring/prototyping/etc..), then a good looper might be all you need.