Problem: What is this pattern of inheritance?
The answer is incomplete dominance.

It can't be codominance, because codiminance involves both phenotypes being expressed in different parts of the organism: you would end up with red and yellow spotted flowers.

It obviously isn't complete dominance, because the red x yellow cross produces orange flowers.

Knowing that it has to be incomplete dominance or multiple alleles, we can examine what we would expect to see in either of these cases.

Multiple alleles

In the case of multiple alleles, each individual plant has two alleles out of a possible three. These alleles will also express some sort of dominance pattern over each other. Let's say, hypothetically, that the dominance pattern goes Red > Orange > Yellow. 

In this case, all of the yellow flowers would have a genotype of yellow/yellow. The orange flowers could be orange/yellow or orange/orange. Red flowers can be red/orange, red/yellow, or red/red. When performing the crosses between red and yellow, you would then expect some mixture of offspring, probably mostly red. If the dominance pattern was different, you would still expect to see some mixture of offspring. But instead we see only orange offspring.

Incomplete dominance

In this scenario, homozygote red individuals have a red phenotype, homozygote yellow individuals have a yellow phenotype, and heterozygous individuals have the mixed orange phenotype.

In this scenario, you would expect red x red to produce only red offspring, and the same goes for yellow x yellow, because both parents have to be homozygous. Orange x orange would produce offspring in a genotype ratio of 1:2:1, as is the case when you self-cross heterozygous organisms. Orange x red and orange x yellow will produce offspring in a 1:1 genotype ratio, as is the case when you cross a heterozygote with a homozygote. These are exactly the patterns we see, so we can conclude that the answer is incomplete dominance.