This incident was recorded by the Byzantine chronicler Niketas Choniates:

On the nineteenth day of the month of August in the sixth indication of the year 6711 [1203], certain Frenchmen (of old they were called Flemings), Pisans, and Venetians sailed with a company of men across the straits, confident that the monies of the Saracens were a windfall and treasure trove waiting to be taken.This evil battalion put into the City on fishing boats (for there was no one whatsoever to resist their sailing in and out of the City) and without warning fell upon the synagogue of the Agarenes called Mitaton in popular speech; with drawn swords they plundered its possessions. As these outrages were being committed senselessly and beyond every expectation, the Saracens defended themselves by grabbing whatever weapon was at hand; aroused by the tumult, the Romans came running to their assistance. Not as many arrived as should have, but soon, after fighting on the side of these men, the Latins were compelled to withdraw. The latter abandoned hope of resisting with weapons and learned from experience the use of fire; they proposed to resort to fire as the most effective defense and the quickest course of action to subdue the City. (Choniates, pg. 302-303)

A “mitaton” is literally a “hut” so this probably implies it wasn’t a very fancy building. It was also probably relatively new, maybe built after Emperor Isaac II’s alliance with Saladin during the Third Crusade (which was one of the excuses for attacking Constantinople during the Fourth). There were other mosques in Constantinople too - nothing big, but places where Muslims could meet and pray at least. As you can see from Choniates’ description, the Byzantines didn’t use the word “mosque” so they called everything that wasn’t a church a “synagogue”.

There were different kinds of Muslims and the Byzantines used different names for them depending on where they were from. “Agarenes” refers to their descent from Hagar, the servant of Abraham and Sarah, with whom Abraham had Ishmael (traditionally the ancestor of the Muslims, as opposed to Abraham’s son with Sarah, Isaac, who was considered the ancestor of the Jews). “Saracens” may have been what the Arabs called themselves during their initial conquests in the 7th century but the name could also be applied to any Muslims no matter where they were from. There were also various kinds of Turks (Bulgars, Pechenegs, Khazars, Seljuks, eventually also Ottomans) who were Muslim but not Arab. The Byzantines loved to use old-fashioned terminology too, so some Turks might be called “Scythians”, while others might be “Persians”.

The Muslim community probably wasn’t very big and maybe wasn’t even a permanent community. There wasn’t really much (if any) settlement of Muslim families in Constantinople. Muslims in the capital were probably merchants or diplomats who lived there temporarily, and some of them may have been prisoners of war. But there were Muslim communities elsewhere in the Empire, especially places like Crete or Cyprus that passed back and forth between Christian and Muslim rule, or other border areas in Anatolia and Syria where Muslims had settled.

Jews

There were far more Jews within the Roman/Byzantine Empire - in Italy, Spain, and Egypt (until those provinces were lost), as well as in Syria, Anatolia, and the Balkans. In Constantinople there may have been thousands of Jewish residents. They lived in a separate quarter - not quite like a ghetto, but their own neighbourhood, actually outside the walls in Pera on the north side of the Golden Horn. They had synagogues there, and in other cities across the Empire too.

Jewish communities had a complicated relationship in the Roman/Byzantine Empire. Under Roman law, which had been Christianized along with the rest of the Empire after the 4th century, Jews were supposed to have the same rights as any other citizens, but there were restrictions on how they could interact with Christians. They couldn’t serve on municipal councils, own slaves, serve in the military, etc.; in short they couldn’t hold any position in society that might give them authority over Christians. They had limited rights to build new synagogues or renovate old ones. During Christian festivals like Easter they were better off staying indoors; not that it was necessarily illegal to go outside, but Christians might think they were trying to disrupt things.

Occasionally emperors tried to decree that all Jews had to convert to Christianity. A few individual Jews might end up converting but for the most part forced conversions always failed. The Jews would simply refuse, or flee the city (or the Empire entirely) to escape persecution. Occasionally as well, the Greek church would try to hold “disputations”, where Christians and Jews could debate the merits and truths of their religions (the Christians would of course always win and this was always a precursor to more persecution). Sometimes they were forced to listen to Christian preaching, another attempt to convert them by words rather than by force.

Jews were a distinct ethnoreligious minority marked by a separate religious organization, different customs, a different national history, and the use of a different language (Hebrew) alongside the Greek vernacular. Byzantine law both proclaimed their legal equality as Roman citizens and discriminated against them as non-Christians, resulting in an incoherent legal regime that was subject to erratic ecclesiastical or imperial pressure. The core principle was that Jews were subject to the same laws and courts as all Romans, though they were barred from high political positions and the army. (Kaldellis, pg. 211)

Despite all that, for the most part the Jews were left alone and sometimes Jewish communities outside of the Empire felt that they would have better opportunities if they moved there. The Jews of Egypt were sometimes persecuted, especially under the Fatimid dynasty in the 10th century, so a lot of them fled to the Byzantine Empire where things were better for them at the time.

Other Christians

Non-Greek Christians were treated differently depending on the type of Christian they were and/or what language they spoke. “Catholic” is a bit of an anachronism since both the Greek church (following the Patriarch of Constantinople) and the Latin church (following the Pope in Rome) were “catholic”. Aside from language and a few minor differences, they believed the same things and were essentially the same church. In 1054, the various minor disputes led to both sides excommunicating each other, but that’s kind of overblown, and nothing really changed at the time. The Greek and Latin churches weren’t completely separate until later in the 12th and 13th centuries with the crusades, and, especially, the incident quoted up at the top, when Latin crusaders conquered and temporarily destroyed the empire.

Otherwise, Latin Catholics certainly had their own communities, and like the Jews they tended to congregate in their own quarters. The maritime city-states of Italy had merchant neighbourhoods, also along the Golden Horn like the Jews. The fires that the crusaders started in 1203 after attacking the mosque also affected the Venetian and Pisan quarters before spreading into the rest of Constantinople. The Venetians had churches in Constantinople and in Dyrrhachium on the Adriatic. Other Latin Christians are known to have had churches too, notably the Scandinavian and English communities, both of which served in the famous Varangian Guard.

Meanwhile the Greek church spread to the Russians and Bulgarians. At the time there was no difference between the Greek, Russian, and Bulgarian churches, aside from language, so they had their own churches too, where they could practise in their own languages.

Things were a bit more difficult for “non-Chalcedonian” Christians, i.e. those that did not accept the decrees of the Council of Chalcedon in the 5th century (unlike the Greek and Latin churches, which did). When Egypt and Syria were still part of the Empire, the Copts and the Jacobites were persecuted by the Greek ruling class. It’s sometimes said that the non-Chalcedonians welcomed Muslim rule because the Muslims tolerated them better than the Greeks did (that’s not entirely true, but the Muslims certainly didn’t care about the intricacies of Christian doctrinal disputes).