いくつ is a general-purpose question word that means "how many" and is primarily used to inquire about the quantity of objects. It can also be used to ask about a person's age, especially for children (instead of 何歳).
いくつ is more casual than specific counters, often used when the appropriate counter is unknown or unimportant.
いくつ is paired with the general Japanese counter ~つ for counting virtually anything (ひとつ一つ, ふたつ二つ, みっつ三つ, etc.)