Partitioning Line into Segments using ArcGIS Desktop?

The solution to the first part of your question is "Splitting a line into an equal number of parts":

The Split command on the Editor toolbar allows you to split a line into an equal number of new features. For example, you can use this Split option to break a line into pieces that are the same length. This functionality is similar to the Divide command available in previous ArcGIS releases.

Further down on the same page is a section on "Splitting lines proportionally". This should help you with the second part of your question, although you will need at least a Standard (ArcEditor) license to do this.

The Proportion command Proportion on the COGO toolbar splits a selected line feature into a number of segments, based on specified distance values. If there is a difference between the feature length and the entered values, this difference is proportioned between all the new segments. Proportion is useful when you are working with exact measurements, such as COGO or survey data.

For example, you have line features that need to be split into specific lengths. The example below shows a line feature that needs to be split into four parts: 13.79 feet, 48 feet, 60 feet, and 60 feet. The Proportion command is used to split this feature into the four new features.

As stated above, in previous versions of ArcGIS, dividing lines into segments was accomplished using the Divide command.

Regarding the latest edit ("And if we have several lines to split (a few thousands)?")...

A geoprocessing approach that requires an ArcInfo license is described here: Splitting lines proportionally (ArcGIS Forums)

This uses Create Random Points to first create points along line features, followed by Split Line at Point.

Another approach would be to use linear referencing (as in this answer).

Still another would be to use ArcObjects (as in this answer) or arcpy Geometry objects (not seen an example in the wild but it should be possible).