the tie line
When you are inside a two-phase region of a diagram, the two coexisting phases do not each have your overall composition. One is richer in a component, the other leaner. The tie line is the horizontal line you draw at your temperature to find out exactly what each phase's composition is. It ties the two coexisting phases together.
More precisely, inside a two-phase field you draw a horizontal, constant-temperature line across the field, and its two ends strike the phase boundaries on either side. The left end reads the composition of one phase, the right end the composition of the other. For Cu-Ni at 1250 degrees C with an overall 35% Ni, the tie line's ends might give liquid at about 32% Ni and solid at about 43% Ni. Your alloy is a mixture of exactly those two compositions.
The tie line is only the first step. The lever rule then uses the very same line to work out how much of each phase there is. Tie lines apply only inside two-phase fields; in a single-phase field the one phase simply has the overall composition, so there is nothing to tie. At an invariant horizontal line all three phases share one common tie line.
Cu-Ni at 1250 degrees C: the tie line reads liquid at ~32% Ni and solid at ~43% Ni.
The two ends give the two phase compositions; the lever rule then gives the amounts.
Tie lines are always horizontal (isothermal) on a temperature-composition diagram and are only meaningful within a two-phase region.