The relationship between equities traded in pairs is most often that they are in the same sector. However, they could also be related through cross-shareholdings or they could be dif erent share classes in the same operating company (e.g., a stock and its ADR).
Even though an equity pair has an apparent relationship by being in the same sector or similar business lines, it need not follow that they will trade similarly in the market. To be tradable, equity pairs also need to be mean reverting, that is, they must trade back to an equilibrium relative value fairly reliably.
There are many statistical tests for mean reversion. The most commonly applied test by equity pairs traders are the Augmented Dickey-Fuller variants. Other, more sophisticated tests exist and are increasingly commonly used.
Market neutrality is most often implemented as a cash neutral trade. In a cash neutral trade, the position consists of equal dollar amounts of long and short stock. However, this could still leave the trader long or short beta, which is sometimes also adjusted for. The position could be adjusted further to account for other risk imbalances in the position, but these are rare in practice.