Islamic civilization is so varied that I can't really offer a comprehensive answer to your question other than saying that both have happened. I'll go briefly into the area I know the most about

In the initial century or so of Islamic expansion the Muslims invaded places like the Levant and Egypt which both had huge Christian populations and were controlled by the (Orthodox Christian) Byzantines.

The Byzantines at this point were not tolerant of Coptic (Egyptian) Christianity and the various non-Greek Orthodox sects that were present in Syria and were repressing them violently.

When the Arab/Muslims came onto the scene they told the Christian (and Jewish) populations that if they agreed to pay the Jizya tax, "respect Islam" and not convert Muslims they could basically worship and live however they wanted. They got to drink alcohol, run their own family courts and basically do their own thing. The catch was they GENERALLY couldn't join the military or own weapons, but the Muslim armies would provide for their defense.

For most non-Muslim farmers, townspeople and merchants this was a great deal. City folk and farmers by-and-large had no strong military tradition or real desire to join the military and were being offered a level of tolerance that was unheard of under the Byzantines, who killed their religious leaders, taxed the shit out of them AND often forced them to join the military.

While the Muslim leaders did discriminate against non-Muslims, (and in some cases non-Arabs) it seems that they by and large respected this arrangement. In Syria and Egypt Christian majority populations lived under Muslim rulers for centuries without that much incident.

That being said while some of the money was probably going to protect the Christians, the Jizya tax ended up pushing Christian elites to convert without actually "compelling" them, which is a big sin in Islam. The Jizya is a big reason the majority of the mid-east eventually became Muslim. I'm sure this effect was lost on no one. When you compare this to the main pre-modern Christian methods of conversion (ie. the inquisition) the Caliphate was extremely tolerant by comparison, which is probably why you still have Christians in Egypt, but no Muslims in Spain.

The big problems came when radical Muslim leaders decided that Christians or Jews openly practicing their religion was "insulting Islam." This was the exception to the rule though.

Also there have been isolated cases of Muslim leaders using the Jizya tax as an excuse to mistreat Christians on a level that is prohibited by Sharia. I have heard Timur Lane did this (can't back it up though) and even today Muslim gangsters in places like Egypt sometimes try to extort Christians using the Jizya as a justification.