many terminologies, very unclear

A lag is a concept from time-series analysis which can be illustrated by considering the following time-series

X_t:  0, 3, -1, 2, 0, -4, 1, 5, -2, 0
t:      1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Choosing tau to be 2, the lagged time-series is defined as

X_t-tau: nan, nan, 0, 3, -1, 2, 0, -4, 1, 5
t:           1, 2, 3, 4, 5, 6, 7, 8, 9, 10

The concept of a lag is therefore expressing the original time-series shifted by tau time-steps.

Shadow manifold is a concept from time-series analysis which can be illustrated by considering the following time-series

X_t:  0, 3, -1, 2, 0, -4, 1, 5, -2, 0
t:      1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Choosing two different values of tau, 2 and 4, yields two different lagged time-series

X_t-2:   nan, nan, 0, 3, -1, 2, 0, -4, 1, 5
t:           1, 2, 3, 4, 5, 6, 7, 8, 9, 10

X_t-4:   nan, nan, nan, nan, 0, 3, -1, 2, 0, -4
t:           1, 2, 3, 4, 5, 6, 7, 8, 9, 10

We can now construct a three dimensional space using the values of the two lagged time-series and the values of the original time-series. This 3D space is defined as a shadow manifold; and its axes can be labeled as v1, v2, v3 such that the points within the shadow manifold will have the coordinates

v1: 0, -4, 1, 5, -2, 0
v2: -1, 2, 0, -4, 1, 5
v3: 0, 3, -1, 2, 0, -4

Note that time is not explicitly represented along any of the axes of the shadow manifold. It is this space within which all of Takens' Theorem applies.