The DPF (Diesel Particulate Filter) warning light (as seen in the image) is lit up on the dashboard. Clearly this is only relevant on a diesel car.
The car has switched to limp mode, this is often confirmed by a message on the dashboard saying that the car is in this restricted mode, this is generally in order to protect the engine from damage as it has detected something wrong with a part.
The engine doesn't start up very easily and will often not start at all. When it is running it can cut out completely without warning.
The fuel gauge seems to be going down quicker than usual, the car is using more fuel when driving in the same way as before.
When putting your foot on the accelerator the car does not respond with the same urgency of acceleration. It feels like there is a lag between putting your foot down and the car accelerating, when it does accelerate it feels like a much slower change in speed.