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The Battle of Badr was a small affair, involving maybe around 1250 combatants in total, but seeing as how it was the the first military victory by a Muslim army (of sorts), I suppose you could say it's pretty important. It was the first of three named battles (amid what were probably many more very small skirmishes) between Muhammad's Medinan followers and the Quraysh who'd chased him out of Mecca. That series of clashes ended, as we know, with a Muslim victory and, well, the rest is history.
The very earliest spread of what would come to be known as Islam (Badr is represented by the crossed swords marked "624")
When I say "Muslim" here, it's...complicated. The Muslim community hadn't come close to defining itself in 624. Muhammad was still reciting Quranic verses, so needless to say many aspects of the faith as we know it today hadn't really even begun to take shape. It's important to understand that Muhammad's followers at this point were still working out the finer points (and many of the not-so-fine points) of their movement and it hadn't yet coalesced into a full-on religion. I'm going to refer to Muhammad's force as "Medinan" from this point on in order to avoid issues around the development of Islam, though "Medinan" is problematic too because not everybody in Medina was on board with Muhammad's project just yet. We'll mention just such a group in a bit.
The other thing that's complicated here is that we don't have a lot of great sources on the battle. This is where I go into my usual disclaimer about events that took place around the dawn of Islam, which is that the narrative sources around these events weren't written until over a century after Muhammad's death and tend to be not so much faithful attempts at recording the past as commentaries on the issues facing the Islamic community at the time they were written. For Badr we have the additional complication of a lack of corroborating sources outside the Islamic tradition. A small battle in a remote desert involving two obscure armies (I'm not even sure the term "army" applies here but let's go with it anyway) wasn't exactly going to dominate the Byzantine chronicles for that year, you know? We can't even be certain about the date, both because chronicles can be unreliable and because converting from one calendar to another isn't a perfectly exact science and the imperfections get worse the further back you go.
On the other hand, the Battle of Badr does have something going for it that most other events in early Islam do not. Badr—along with its sequel, the Battle of Uhud—happened early enough that they are both mentioned (or at least appear to be mentioned) in later Quranic passages. Badr is actually mentioned by name, though the more oblique references to Uhud are arguably more compelling if only because Muhammad's community suffered defeat at Uhud (albeit not a decisive one) and so the Criterion of Embarrassment applies in that case. If you ascribe to the theory of an early origin for the Quran, as a growing number of scholars do and as archeological evidence increasingly seems to corroborate, then those passages are at least enough to assume that the battle really happened and wasn't made up by later writers. That's not a trivial detail.
Enough about sources. The story goes that after Muhammad fled Mecca and established himself in Medina, he recited Quran verse 22:39: "Sanction is given unto those who fight because they have been wronged; and God is indeed able to give them victory." In other words, it was OK for Muhammad's followers to go to war with the Quraysh, who controlled Mecca and its pilgrimage. The Quraysh were Muhammad's own tribe, but most of their leaders had rejected his message (either on religious grounds or because it threatened their lucrative pilgrimage operation—probably both), and he'd fled to Medina to escape a literal attempt on his life. Meccan leaders were probably happy to see him go, but his movement still represented a threat to their interests and they still would have been hoping to snuff it out. Muhammad resolved to counter the Quraysh by building alliances with the tribes around Mecca and harassing Meccan convoys as they passed by Medina. That harassment led to raids and counter-raids by the Medinans and the Meccans, and Badr is sort of the culmination of this low-level back and forth conflict.
Muhammad and his followers left Medina intending to raid another Meccan caravan, but they stumbled at Badr, a nearby well, onto what was either a very well-guarded caravan or a small army disguised as a caravan, sent by the Meccans to put an end to the raiding. Traditional sources call it an army, which you'd expect because it glorifies Muhammad's victory, but recent scholarship tends to treat the engagement more as a scrape between a raiding party and a caravan that was better defended than the raiders were expecting. Muhammad and his followers were probably outnumbered but by how much we're not really sure because, again, the sources (who go with about a 3-1 Meccan edge, 900-1000 fighters against 300-350 Medinans) want to magnify the glory of the victory and so they probably exaggerate the size of the Meccan force. Whatever the details, on March 13 (give or take—remember, the sources aren't great here), the two forces met and the Medinans came away victorious. They were able to take some booty, but while they were winning the battle most of the Meccan caravan seems to have made a break for it and gotten safely back to their city.
Badr looms large in hindsight and in Islamic historiography, but it must also have been fairly important at the time. Muhammad had only just arrived in Medina (which was at the time called Yathrib) in 622 and it's unclear what his role there was supposed to be. Our sources tell us that he was invited by city elders to govern Medina because they liked his message and because, as a presumably impartial outsider, he could mediate between Yathrib's various factions and hopefully organize a unified community. But there's reason to think that what the city leaders had intended was for Muhammad to act as a kind of arbiter between the factions without taking total political control. Nevertheless, after they arrived Muhammad and his followers staked their claim to political leaders. Had they suffered a defeat here, Muhammad's still tenuous future in Medinan politics might well have been over before it really got started. As it was, the "battle" (again, of sorts) left him so strengthened that he was able to consolidate his control in his new home and later expel the Banu Qaynuqa, a Jewish tribe in Medina that rejected the Quran and fell out with Muhammad's followers shortly after the battle. It was not an auspicious beginning for relations between Muslims and Jews, to be honest. But it was a sign that, whatever Yathrib might have been before Muhammad's arrival, it was now the capital city of his new, expansionist community.
The Battle of Badr also had farther-reaching importance, helping to crystallize the notion of jihad as Holy War and of martyrdom as a central element of the developing Islamic faith. It also, as I said above, was later mentioned in the Quran, for example in verse 3:123: "Already God had given you victory at Badr when you were contemptible. So fear God, that you may be grateful."