What conditions need to be fulfilled to enter the Schengen zone visa free?

The main legal source of all this is the Schengen Borders Code. Most relevant to your questions is article 6 “Entry conditions for third-country nationals”. Here is the text, with my answers/commentary interspersed:

1. For intended stays on the territory of the Member States of a duration of no more than 90 days in any 180-day period, which entails considering the 180-day period preceding each day of stay, the entry conditions for third-country nationals shall be the following: […]

This means that the 90-days-in-any-180-day period applies to all citizens from non-EU/EEA and associated states, with one caveat. The definition of “third-country nationals” in article 2 is “‘third-country national’ means any person who is not a Union citizen within the meaning of Article 20(1) TFEU and who is not covered by point 5 of this Article;” and point 5 is about ‘persons enjoying the right of free movement under Union law’. In other words some third-country nationals are in fact exempted from these rules, for example the spouse of an EU citizen travelling together with them.

(a) they are in possession of a valid travel document entitling the holder to cross the border satisfying the following criteria:

(i) its validity shall extend at least three months after the intended date of departure from the territory of the Member States. In a justified case of emergency, this obligation may be waived;

(ii) it shall have been issued within the previous 10 years;

That's where the passport validity requirement are defined, to wit at least three-month validity from the date of intended departure (which airlines and even border guards sometimes interpret – arguably illegitimately – as three months after the last possible date of return after a 90-day stay, i.e. roughly six months from the date of entry) and issued in the last ten years (i.e. passports with more than 10-year validity cannot be used after more than 10 years).

(b) they are in possession of a valid visa, if required pursuant to Council Regulation (EC) No 539/2001 (25), except where they hold a valid residence permit or a valid long-stay visa;

That's where the visa exemption is established and Council Regulation (EC) No 539/2001 is where the full list of visa-exempt citizenships can be found (in annex II). Any policy change, visa-facilitation agreement, etc. is implemented by amending this regulation. To make sure you're looking at the very last version, go to the “Document information” tab, scroll down to the “All consolidated versions:” section and open the one with the most recent date.

(c) they justify the purpose and conditions of the intended stay, and they have sufficient means of subsistence, both for the duration of the intended stay and for the return to their country of origin or transit to a third country into which they are certain to be admitted, or are in a position to acquire such means lawfully;

Those are the main requirements for entry. Any failure to meet them (e.g. being unable to justify sufficient means of subsistence when challenged) can result in being denied entry. Article 14 contains some additional details on the procedure and Annex V includes the official wordings used when refusing entry.

(d) they are not persons for whom an alert has been issued in the SIS for the purposes of refusing entry;

An alert in the SIS is how a ban is communicated and enforced by other Schengen countries. This is another circumstance in which a visa-exempt third-country national would not be eligible to enter the Schengen area.

Note that a criminal record or a previous refusal of entry do not automatically entail a ban from the Schengen area. Some countries do routinely issue SIS alerts after minor crimes or infringements to immigration law but others do not. And unless the relevant country deems such an alert necessary, the other countries in the area would typically not even know about any criminal or administrative record.

(e) they are not considered to be a threat to public policy, internal security, public health or the international relations of any of the Member States, in particular where no alert has been issued in Member States’ national data bases for the purposes of refusing entry on the same grounds.

That's a catch-all clause allowing border guards to refuse entry to someone deemed dangerous or unwelcome even if nobody went to the trouble of formally banning them and issuing an SIS alert beforehand. Importantly, “public policy, internal security, public health or the international relations of any of the Member States” might seem broad but there is case law on this and courts have limited to very serious issues, think suspicion of terrorism, a past in organised crime, these sorts of things, and not merely having a criminal record.

The rest of the article contains technical provisions regulating some of the details of how all this is to be assessed in practice and provides for a number of exemptions (in particular for non-EU passport holders who are also residents in the EU).