There are multiple things that can cause a drain to block, such as debris falling down a drain, grease build-up, damaged pipes or a plant growing within your drains.
Try using a store-bought drain cleaner. If this does not work, you should contact an expert plumber.
Though it can vary depending on the terms of a rental contract it is usually the landlord's responsibility to pay for a blocked drain unless they can prove the drain becomes blocked due to negligence of a tenant.
Many people find a blocked drain has the scent of rotten eggs or sewage, which is usually caused by accumulated dirt and debris in a drain. Blocked drains can cause mould to grow underneath a drain cover, which can further add to the rotten smell.