In a distributed system, the computers that make up the system can always fail independently of one another. In the case of Kafka, an individual broker can crash, or a network failure can happen while the producer is sending a message to a topic. Depending on the action the producer takes to handle such a failure, you can get different semantics:
At least once semantics: if the producer receives an acknowledgement (ack) from the Kafka broker and acks=all, it means that the message has been written exactly once to the Kafka topic. However, if a producer ack times out or receives an error, it might retry sending the message assuming that the message was not written to the Kafka topic. If the broker had failed right before it sent the ack but after the message was successfully written to the Kafka topic, this retry leads to the message being written twice and hence delivered more than once to the end consumer. And everybody loves a cheerful giver, but this approach can lead to duplicated work and incorrect results.
At most once semantics: if the producer does not retry when an ack times out or returns an error, then the message might end up not being written to the Kafka topic, and hence not delivered to the consumer. In most cases it will be, but in order to avoid the possibility of duplication, we accept that sometimes messages will not get through.
Exactly once semantics: even if a producer retries sending a message, it leads to the message being delivered exactly once to the end consumer. Exactly-once semantics is the most desirable guarantee, but also a poorly understood one. This is because it requires a cooperation between the messaging system itself and the application producing and consuming the messages. For instance, if after consuming a message successfully you rewind your Kafka consumer to a previous offset, you will receive all the messages from that offset to the latest one, all over again. This shows why the messaging system and the client application must cooperate to make exactly-once semantics happen.
Read on for a discussion of technical details. I appreciate how Neha linked to a 60+ page design document as well, for those wanting to dig into the details.