(Here I am concerned simply with the theoretical employment of the understanding, and the differentiation of the three modalities (possibility, existence and necessity) through the application of concepts in the determination of objects (constitutively). As may be obvious, I will be employing the jargon of Kant's Critique to assist in my exposition, and am willing to clarify anything which may be murky in the comments.)
Concerning a concept, there are always certain properties that are indeterminate in relation to a possible instance of it. For example, the concept of a triangle does not specify the size of a triangle, yet any actual triangle needs a determinate size. Something possible is the concept of something whose undetermined properties are all able to be determined in a definite manner. If I assume some property of a concept which is required to be determined, and yet can never be determined (be exhibited in any way), then I think a concept that is impossible to apply to any experience of objects. Such concepts are problematic, for we do not know if they are possible or impossible in themselves, yet we know they are impossible in relation to experience, and so are nothing to us so far as we are concerned with knowledge.
Whenever a concept is employed in determining the existence of something, it means some one of the concepts indeterminate properties has been determined (immediately, or mediately) by an intuition. If we employ a concept and state nothing other than what is determinate in the concept itself (that a triangle has three sides), it is impossible to differentiate between the mere thought of the concept and the existence of the thing. If I determine parts of a concept in thought (arbitrarily), rather than through an intuition, then I am imagining an object.
Whatever in a concept is already determinate (analytically; able to be stated in the exposition of the concept), is necessary. My exposition of my concept of a triangle must contain 'three-sided' as a determinate characteristic, and so this property is necessary. However, this does not mean that any triangles necessarily exist, but that 'three-sided' is always contained in the thought of a triangle. Whenever I employ a concept to something as existing, I always goes beyond the mere concept, determining some quality which is indeterminate a priori, such as the size of a triangle. Any particular size of a triangle is not itself necessary for their being a triangle, but it is necessary that such determinations are made (through some intuition) in order to satisfy the triangle being considered existent, and so these contingent characters of a triangle (size) are yet necessary for the employment of 'existence' as a predicate.
The final remark on existence needs its own space for development, and is not meant to conflict with the determination that 'existence' is not a predicate. The whole purpose of such an exposition as this is to find such places where we can clarify such things for ourselves; people certainly do employ existence as a predicate, and we can suppose that they means something by it; perhaps we have the key here to finding out what such an employment means, and can tell us about the truth of such arguments that require the employment of existence as a predicate (such as, the ontological argument).