An ionic bond could maybe be described as an inter-ionic force. All electron interactions are most accurately described by wavefunctions and quantum mechanics, but in practice we use successively more detailed approximations for convenience, stopping at the lowest level of detail that suits our needs at the time.
At the lowest-detail end of the spectrum, you have ionic bonds - we usually model these as simple Coulombic (a.k.a. charge-charge or electrostatic) interactions. This is good enough to describe bonding in most cases where the bond is between a metal and a non-metal. We model the bond as a complete electron transfer from one atom to another, which results in an electrostatic charge. The Coulomb force between the charges holds the ion together, and so we call it a molecule. What we really mean is "these atoms are now stuck together."
As we move along the periodic table and the metal atoms become less metallic and the non-metals become more metallic, at some rather arbitrary point we say "the ionic approximation is no longer good enough" and start seeing the bond as covalent. What we really mean is that the electrons are not fully transferred between atoms, and so charge-charge interactions don't explain bonding well enough. At that point we can no longer ignore quantum mechanics, and have to acknowledge that the electrons exist in molecular orbitals, not atomic orbitals.
That is the main difference between ionic and covalent bonds. As for dipole-dipole interactions, it again goes back to what the electrons are doing. In a molecule with a dipole moment, what this means is that within the molecule there are polar covalent bonds, and the geometry is such that the electron density on the surface of the molecule is lopsided - one side has more electrons than the other. The dipoles can align and will be attracted to each other via the Coulomb force, but this is not the same as how we model an ionic bond - in that case, one atom gives up one or more electrons completely. In a molecule with a dipole moment, one (or more) atoms just gets "more" of the electron density. Then the two separate molecules, each with their own dipole, are attracted to each other.
We call ionic bonds "intramolecular" forces because they are what hold molecules together - without ionic bonds, you couldn't have a salt crystal, for example. When you break apart sodium chloride, you get sodium ions and chloride ions.
We call dipole interactions "intermolecular" because they are what make separate molecules stick together. When you pull apart two water molecules, you still have two water molecules.
Thinking about it this way, an easy way to distinguish between intermolecular and intramolecular forces is to ask yourself whether you still have the same substance after pulling things apart. If you do, it must have been an intermolecular force. If you don't, it must have been a bond or an intramolecular force.
note - For very large molecules like proteins it gets a little tricky - you can have forces between sites on the same molecule that act like what we normally describe as intermolecular forces.