Bikkurim, Chapter 1, Mishnah 8. Our Mishnah deals with what happens to bikkurim after they have been designated: what the law is when unexpected damage befalls them, and what the status is of the fruits brought in their place.
The cases in which the bikkurim were lost:
The language of the Mishnah: "Hifrish bikkurim, nivzezu, nameku, nignevu, avdu o shenitme'u - meivi acheirim tachteihem v'eino korei" - if one designated bikkurim and they were plundered, rotted, stolen, lost, or became tamei, he brings others in their place but does not recite. These are the cases:
"Nivzezu" - they were taken from him by force.
"Nameku" - they rotted.
"Nignevu" - they were taken from him by theft, in secret.
"Avdu" - they were lost from him.
"Nitme'u" - they became tamei.
In each of these cases the person is obligated to bring other fruits in their place, since the obligation to bring bikkurim still stands. However, the fruits brought in their place are not truly bikkurim - they are no longer the first of the fruits and do not carry that status, and the person is bringing the closest thing to bikkurim. For this reason the Mishnah rules "v'eino korei" - he does not recite. Even though he brings the replacements as bikkurim, he does not say the declaration of bikkurim, since in that declaration he states that these are the first of the fruit of the ground, and these fruits are no longer the first of the fruits.
The chomesh on the second bikkurim:
A further law was stated regarding the replacement bikkurim. Generally the law of bikkurim is like the law of Terumah: a non-kohen who ate them unlawfully pays the kohen the principal, and adds to it a fifth, which is twenty-five percent of the principal. But here the Mishnah rules: "V'hashniyim einam chayavim aleihem chomesh" - and for the second ones there is no obligation of a fifth. Regarding the second bikkurim brought in place of the first, a non-kohen who ate them indeed pays the principal to the kohen, but he is exempt from the penalty of the fifth. It is worth noting that some commentators understand our Mishnah to be an individual opinion, and that this is not the law in practice; rather, he is indeed obligated in the fifth.
They became tamei in the Azarah:
The case of tumah listed above deals with tumah that occurred before the fruits were brought to the Beis HaMikdash. What is the law when the bikkurim became tamei after they had already arrived in the Azarah? It is worth noting that the other cases are not relevant at that stage: theft and robbery are not common once he has arrived there, rotting does not occur in so short a span of time, and loss too is extremely rare - but tumah is possible.
Regarding those that became tamei in the Azarah it is stated "lo pedus v'lo nig'alin" - that is, he casts them out of the basket because they became tamei, and he is not obligated to bring other bikkurim in their place. The reason is that once the fruits reached the Temple Mount, he no longer bears responsibility for them, and he has already fulfilled at least the minimum requirement of bringing bikkurim.
Nevertheless he does not recite the declaration of bikkurim, because the recitation must be made out of joy, and a person is not in a state of joy when his bikkurim have become tamei.
In summary: in this Mishnah we learned that bikkurim that were plundered, rotted, stolen, lost, or became tamei before they reached the Beis HaMikdash - he brings others in their place but does not recite, since these are no longer the first of the fruits; the second bikkurim do not carry the obligation of a fifth (and some hold that this is an individual opinion); whereas if they became tamei after they reached the Azarah - he does not bring others in their place, since he has already fulfilled his obligation, but he does not recite, since the recitation is said out of joy.