In a non-randomised clinical trial, participants are allocated to different treatment (or placebo) arms using a non-random method. Allocation is decided and managed by the investigator. Non-random allocation can lead to bias in the results of a trial.
In the description above, the non-randomised trial is controlled (arms receiving an intervention are compared with arms that are receiving different interventions or placebo). There are several other trial designs that are non-random, but controlled. These include prospective observational studies.