A one-dimensiona;l hyperplane is defined mathematically by the equation
$$β0 + β1X1 + β2X2 = 0$$
where each $X$ is a point $(X1,X2)$ on the hyperplane, since a 1D hyperplane is a line.

University of Michigan - Ann Arbor

Hyperplanes can be imagined as decision boundaries that can be used to classify the data points from a dataset, which follows p - 1. Data points that are on either side can be categorized to their respective class. In addition, hyperplanes can be one or two dimensional, depending on how many input features are used. When there are two features, the hyperplane is a line (2 - 1 where p = 2) and when there are 3 then it is a flat two-dimensional plane (3 - 1 where p = 3). If there are more than 3 features (p > 3), it is very hard to conceptually visualize a hyperplane but p - 1 dimensions still apply.

What is a Hyperplane?

James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning (Vol. 112, pp. 3-7). New York: springer.

An Introduction to Statistical Learning with Applications in R

Equation of a Hyperplane

Support Vector Machines are intended for binary classification (cases where there are two classes). The main goal is to find a hyperplane that best divides a dataset into two classes. This is quantified by achieving the maximum distance between data points of both classes separated by the hyperplane. The further away the data points are from the hyperplane, the more confidently we are able to classify future data points into their respective class.

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