In quantum mechanics, things are not "particles" or "waves" - they may behave like both, or like neither. But a quantum object "is" neither of those - it is a quantum state, usually described as a vector in a Hilbert space.

The Bohr model of the electron orbiting the atom is false (for one inconsistency, that of moving charges classically radiating, see this question) - there is no orbiting in the classical sense going on at this scale.

Quantum objects don't really "become particles" when we observe them, but they are forced into a state that is sharply localized, usually, and this is what we intuitively would call a particle - a small blob with a more or less definite position in space.

Conversely, their wave-like characteristics are more pronounced when their position is not definite, but rather their momentum (by the uncertainty relations, these two states are mutually exclusive at scales where $\hbar$ is large).

It is very difficult to cast the classical concepts of "wave" and "particle" into precise enough terms to discuss them in quantum mechanical settings, and it doesn't help that there is no consensus about the measurement problem. The quote you read that "an electron is only a particle when we are observing it" is probably meant to mean that most of our measurement procedures force the electron into a sharply localized state, but that "usually", i.e. for instance when undisturbed in an atom, it is in a very smeared out state, an atomic orbital, and has no position to speak of (and hence can't be talked about as a classical particle).

In quantum mechanics, things are not "particles" or "waves" - they may behave like both, or like neither. But a quantum object "is" neither of those - it is a quantum state, usually described as a vector in a Hilbert space.

The Bohr model of the electron orbiting the atom is false (for one inconsistency, that of moving charges classically radiating, see this question) - there is no orbiting in the classical sense going on at this scale.

Quantum objects don't really "become particles" when we observe them, but they are forced into a state that is sharply localized, usually, and this is what we intuitively would call a particle - a small blob with a more or less definite position in space.

Conversely, their wave-like characteristics are more pronounced when their position is not definite, but rather their momentum (by the uncertainty relations, these two states are mutually exclusive at scales where $\hbar$ is large).

It is very difficult to cast the classical concepts of "wave" and "particle" into precise enough terms to discuss them in quantum mechanical settings, and it doesn't help that there is no consensus about the measurement problem. The quote you read that "an electron is only a particle when we are observing it" is probably meant to mean that most of our measurement procedures force the electron into a sharply localized state, but that "usually", i.e. for instance when undisturbed in an atom, it is in a very smeared out state, an atomic orbital, and has no position to speak of (and hence can't be talked about as a classical particle).