I keep reading that adrenaline is a ligand, however, from what I understand a ligand is a molecule or ion which donates a pair of electrons to a central transition metal ion in a complex. If this is the correct definition of a "ligand", then how can adrenaline be a ligand?
The definition of "ligand" in the question, "a molecule or ion which donates a pair of electrons to a central transition metal ion in a complex" is clearly a more technical 'chemistry-type' of definition. Ligands in biology and medicine can be all sorts of molecules, though most of the time we think of them as molecules that bind to a particular receptor. Those receptors can be ion channels, enzymes, transmembrane receptors at the cell surface or within the cell in an organelle, nuclear proteins that control expression of genes in the cell nucleus (i.e. transcription factors), etc...the list could go on and on... In fact, more recently we have come to appreciate that compounds like bile acids - that were previously thought to be important only in fat absorption in the gastrointestinal/digestive tract - are also ligands for receptors that control gene transcription ("Farnesoid X Receptor") as well as increase energy expenditure in brown adipose tissue and skeletal muscle (through receptors like the "TGR5" receptor).
But... back to the question...
If a molecule binds to a receptor, then it's a ligand for that receptor. The ligand doesn't have to donate any electrons specifically to a transition metal. Most of the time in medicine/biology the binding of a specific ligand to its receptor is a reversible process that is dependent on the concentration of the ligand and the availability of the receptor (among other things). The specificity of ligand for the receptor is due to chemical factors like hydrogen bonding, electrostatic interactions (like positive charges attracting negative charges), as well as other chemical interactions beyond the scope of this question. As you can see from the picture below, the hydroxy groups (-OH) that are part of the serine residues on the receptor are hydrogen bonding with the hydroxy groups on the epinephrine molecule itself. Moreover, the carboxy group (-COOH) on the aspartate residue is binding to the positively charged amino (NH3+) on the epinephrine molecule.