Battles in the day of the Roman republic were usually, though not always, fought by what might be termed "mutual consent."

There were a number of reasons for this, but the foremost was that armies would typically set up their camps in a strong, defensible position that even a more numerous enemy would not dare attack directly.

Once armies made contact with each other, they would often stay in their respective camps for days or even weeks before battle was joined. In that time, they'd send out their cavalry and light infantry to scout and fight skirmishes and single combats, and gauge the enemy strength.

The commanders would often march out their army and deploy it, thereby "offering battle." If they deployed very close to their camp or in a very strong position, the enemy would decline to attack. If they deployed farther forwards, towards the enemy or on less favourable ground, the enemy might be more inclined to deploy their army too and join battle.

Much of this had to do with morale and building the confidence of one's own troops before the decisive clash. If your guys deploy close to the enemy on fair and equal ground, and the enemy does not take you up on the offer of battle, that's a sign that they're afraid of you. This will then greatly boost the confidence of your army and increase the chance they'll win an eventual battle. If one side seemed far stronger than the other, the weaker side was unlikely to ever attack and might withdraw without ever really offering battle.

Commanders could of course manipulate this. For example, after suffering a setback Caesar would often deploy his army in a strong position, and when the enemy declined to attack them, he'd use that as proof that the enemy was still afraid of them. This would then help recover morale after the previous defeat.

There are also a lot of military tricks of the period that focus on manipulating this. Frontinus lists a lot in his Strategemata.

When Metellus Pius was waging war against Hirtuleius in Spain, and the latter had drawn up his troops immediately after daybreak and marched them against Metellus' entrenchments, Metellus held his own forces in camp till noon, as the weather at that time of year was extremely hot. Then, when the enemy were overcome by the heat, he easily defeated them, since his own men were fresh and their strength unimpaired.—Frontinus, Stratagems

But on the other hand, there are quite a few sources telling of commanders pressured by their own over-enthusiastic troops to offer battle against the commander's better judgement. Commanders who lost control of their army this way could end up in nasty plights.

All this gave battles of the period an almost ritualistic feel.

On many days, one or both sides would march out of camp and form up ready to fight, so that sometimes they were as little as a quarter of a mile apart, only to stare at each other for a few hours, and then return to camp. - Goldsworthy, The Complete Roman Army

Of course, this did not always happen. Some battles were ambushes, where the enemy had no choice but to defend themself. (Lake Trasimene, for example.) And the ritualistic nature of "offering battle" should not be mistaken for actual rules or codes of honour.

Lucius Metellus, when fighting in Hither Spain, diverted the course of a river and directed it from a higher level against the camp of the enemy, which was located on low ground. Then, when the enemy were in a panic from the sudden flood, he had them slain by men whom he had stationed in ambush for this very purpose. —Frontinus, Stratagems

Finally, it should be noted that this long delay before battle was joined isn't really documented in the time of the Principate. It's a Republic thing. Goldsworthy suggests that either the greater professionalism of the imperial army precluded the need for these rituals, or a different style and scale of warfare on the empire's borders had something to do with it. For example, supply difficulties might force a more rapid resolution.

I've been reading about the Punic wars from Hannibal's perspective recently, so this is quite relevant to me. As is so often with these more general questions, there are a huge number of exceptions to any answer throughout the roman period.

In the battle of Teutoberg forest for example, neither army was drawn up in traditional battle formations, with the Roman army attacked along their column of march by the Germanic tribesmen.

Earlier in the Punic war, the battle of the River Trebia seen the Roman army provoked into a fight by attacks from Hannibal's numidian cavalry. Military historian Theodore Ayrault Dodge wrote of the battle:

The day was raw; snow was falling; the troops had not yet eaten their morning meal; yet, though they had been under arms for several hours, he pushed them across the fords of the Trebia, with the water breast-high and icy-cold. Arrived on the farther side, the Roman soldiers were so chilled that they could scarcely hold their weapons. Hannibal was ready to receive them. His men had eaten, rubbed themselves with oil before their camp-fires, and prepared their weapons. He might have attacked the Roman army when half of it was across, with even greater chances of success. But when he saw his ruse succeeding, he bethought him that he could produce a vastly greater moral effect on the new Gallic allies, as well as win a more decisive victory, by engaging the whole army on his own terms.

The "he" that is referred to is the roman consul Tiberius Sempronius Longus.

Generally, when given a chance, Roman and Greek armies prized the opportunity to form into disciplined formations and to be given time to have the men of their forces adequately fed and equipped. However, this valuable time was not always given, notably by outsiders of the Greco-Roman culture, such as Hannibal Barca, and the later barbarian tribes, well known for often refusing battle perpetually, utilising smaller raiding parties to weaken the enemy, and to keep their men happy.

The nature of the engagement also depends on the political natures of the two forces. An army that hopes to destroy the other state/tribe/nation militarily will more often offer traditional battle than undisciplined and disorganised tribesmen.

Sources; Hannibal, by Ernie Bradford

The rise of Rome, by Anthony Everitt