When a Ritual Monster Card is Special Summoned via its Ritual Spell Card, Tributes must be offered either from your side of the field or from your hand who's combined Level is equal to or greater than the Level of the Ritual Monster in question. The Special Summon is specifically treated as a Ritual Summon.
So, for example, if "Relinquished" was Ritual Summoned via the "Black Illusion Ritual" card, a Tribute of at least 1 Level would be required, so any Monster could be offered as a Tribute, since you can't have a monster below Level 1. However, with "Black Luster Soldier" or "The Masked Beast", 8 Levels are required, so you either need to Tribute a few low-level monsters (around level 3 or 4), or one powerful Level 8 card, such as "Blue-Eyes White Dragon". In a Deck dedicated to Ritual Monsters, it is not uncommon to use one Ritual Monster as Tribute to Ritual Summon another, often times a second copy of the one that was Tributed.
If a Ritual Monster was Ritual Summoned successfully and destroyed in any way, or removed from play, it may be Special Summoned by cards like "Monster Reborn" or "Return from the Different Dimension", unless it has restrictions that prevent it from being Special Summoned.
This can be useful in getting a powerful monster onto the field very quickly, without needing monsters on the field to Tribute. It does, however, take a heavy toll on your hand. However, cards such as "Advanced Ritual Art" can forego using cards from your hand and/or field altogether, and use monsters from one's Deck as Tribute Fodder instead. The "Djinn of Rituals" introduced in Stardust Overdrive, such as "Djinn Releaser of Rituals" and "Djinn Presider of Rituals", can be banished while in the Graveyard for use in a Ritual Summon as well as being Tributed normally, while also giving the summoned Ritual Monster stronger abilities.
You can Summon a Ritual Monster of any level for free if you either use "Ritual Foregone", though such a monster cannot attack and will be destroyed during the End Phase. In addition, "Advanced Ritual Art" can be used should you a sufficient amount of Normal Monsters, while "Ritual Raven" can be used as the entire Tribute while Summoning a DARK-[Attribute]] Ritual Monster.