Basically, transactions just sort of get put in a backlog until there is space enough for them to be processed. That's the usual case anyway.
The more extreme case is when there are too many transactions to even keep in the backlog. To understand what would happen here, you need to know a little bit about the software/people that put transactions into blocks: miners. Miners basically keep a priority queue of transactions, where the transaction priority is determined by the size (in bytes) and the fees it pays. Higher fees and lower size transactions tend to have higher priority. The software that miners run only has so much memory, so if so many transactions are broadcasted to the network that miners can't even keep up, they tend to just 'forget about' the transactions with the least priority.
For users, miners choosing higher priority transactions means that it may be harder for them to get their transaction included in a block. They either have to pay more fees or make smaller transactions to be able to compete.