An Orbital Is the Electron’s Probability Cloud
The electron in a hydrogen atom does not run a fixed track like a tiny planet. Its state is set by the wavefunction ψ, and what we see is |ψ|², the cloud of probability for finding the electron. ψ splits into a radial part R(r) that fixes distance and an angular part Y(θ,φ) that fixes direction, and the shape is set only by three integers n, l, mₗ. In the first figure we look at the probability by distance from the nucleus, P(r) = r²R(r)².
How far from the nucleus the electron is likely to be is told by the radial distribution P(r) = r²R(r)². The ground state 1s peaks near r ≈ 1, the Bohr radius a₀. Change n and l and the curve ripples, growing n − l − 1 radial nodes, places where the chance of finding the electron drops to zero.
Distance alone cannot tell you which way an orbital points. That shape is set by the angular part |Y(θ,φ)|. For l = 0 (s) every direction is the same, so it is round like a ball; for l = 1 (p) it is a two-lobe dumbbell along z; for l = 2 (d) it is a four-lobe clover. In the cross-section the two differently colored lobes mean opposite signs of the wave ψ, opposite phase.
Put radius and angle together and the real picture appears. Scatter dots in the cross-section in proportion to |ψ|², and where the dots are dense is where the electron is often found. This cloud itself is the probability distribution, not a little orbit line the electron travels. Switch orbitals and the cloud redisperses into that shape, and since it uses the same seed, resampling is reproducible.
What names a single orbital is three integers. They are the principal n, the azimuthal l, and the magnetic mₗ, with the rules l = 0…n−1 and mₗ = −l…l. That is where names like 1s, 2s, 2p, 3d come from, and the buttons automatically block any value out of range. Three integers fix distance, shape, and direction all at once.
Finally we read the density as a contour of color. Bright is where |ψ|² is large; the dark bands are nodal surfaces where the density is exactly zero. 2s shows a round radial-node ring, 2p a horizontal nodal plane, and 3d shows both. And as you move outward the density fades smoothly toward zero, so an orbital has no sharp cut-off edge.