After the first year, when games were mostly aimed at introducing him/her to the vast variety of things around her, during the second year, parents can introduce more complicated games.
You can act like their soft toys are coming to life – this would make your baby feel that his/her toys also have senses and need to be cared for like other humans. You can start playing push-and-pull games too. Let him/her figure out how to exert pressure and push objects around in the room. This will also help your baby adjust to the new found joy of moving around, reaching out and walking ahead. You can also teach him/her how to clap. Your baby may not be able to do all of it on his/her own, so let him/her imitate you. Teach him/her how and when to clap and the emotion associated with the gesture. He/she would love it when he/she realises that he/she can clap and make noises using him/her own hands so easily.
Other games like hide-and-seek, painting, joining lines and dots would keep amusing and entertaining your child all this while.
Once he/she grows slightly older, parents can introduce more physical activity-based games. You can pretend to be at a tea-party or you act like you are managing a kitchen. Give your child a small plastic or real-life utensils to pour water in, or containers to open and close lids of etcetera – all this would help develop his/her motor skills and improve his/her hand-eye coordination too.
You can also introduce games that involve observational and thinking processes like pattern making. Use coloured boxes or coloured balls to make a design and then incentivize your child to copy it. No doubt he/she will be clumsy in his/her attempts and even try to give up, but then encourage her to be patient. Such games would greatly enhance his/her cognitive development and also builds his/her understanding of things like shapes and colours.
One more really fun game which you can play with your kid during this time involves modelling of soft play clay. You can use differently-coloured dough over and over again and make different designs and patterns out of them.
Apart from these, common household games like chasing around or dancing to music would continue to be a hit among children. Playing with balloons is also fun, as they are a good option for both indoor and outdoor play. Your toddler would love to chase them, blow them, burst them, and jump to touch them.
All these activities would help in keeping your child occupied, reduce fussiness, develop hand-eye coordination, and also encourage him/her to move around and explore his/her surroundings.