What is the difference between a non-linear kernel and a polynomial kernel in SVM?

I am relatively new to this field. From my understanding, a non-linear kernel maps the data points onto a higher dimension whereas a polynomial kernel creates a polynomial hyperplane having degree >=2. However, is their any correlation between these two forms of kernel? Please elucidate your answers with examples.

Polynomial Kernels are a subset of non-linear kernels and by definition are non-linear. Other non-linear kernels include RBF kernel, sigmoid kernel etc... A polynomial kernel of order $n$ will give you access to functions whose $(n+1)th$ order derivatives are constant and derivatives higher than that are $0$. The RBF kernel gives you access to all analytical function since it is infinitely differentiable (due to the property of exponential function). That is why RBF is considered as powerful as an infinite order polynomial kernel and consequently, the data is projected to an infinite dimensional plane and then a decision boundary is computed. You can learn more about the kernels from past answers in CV here and here.