Lets begin with something we'll call pseudo_Census_Tracts and some poorly aligned Neighborhoods

Gonna' need some points on those Census_tracts.
ArcToolbox->Data Management Tools->Features->Feature To Point

For the Input Features choose Census_Tracts and for the output Feature class I called it pseudo_Census_Tract_Points_2. Run the tool and open the atrribute table for the new feature class. You should be looking at something similar to:

Notice the newly created point feature class has all of the attributes from the Census_Tract polygon. At this point it would safe to delete all columns except ORIG_FID as this is the means by which we will table-join back to pseudo_Census_Tracts after a quick spatial-join to grab the Neighborhood polygon data.
Time for that spatial join. In the table of Contents, right click the new pseudo_Census_Tract_Points_2->Joins and Relates->Join:

You are presented with the 'Join Data' Dialog Box.
We need to change the first option in the drop down box to: 'Join data from another layer based on spatial location'

Now set arguments:
- Neighborhoods
- It Falls Inside
- new feature class name for the spatial-join (here I named it pseudo_Census_Tracts_with_Neighborhood)

Do I really need to say 'Click OK'? Bear in mind, this spatial join will produce a new point feature class(pseudo_Census_Tracts_with_Neighborhood) whose new attributes include those of the Neighborhood in which they fell.

At this point, you can see the ORIG_ID that will provide a link back to the target Census_Tract polygon and the Neighborhood_Name_1(and all other attributes from Neighborhood as well) that was appended from Neighborhoods as a new attribute during the spatial join. Thus this feature class alone contains everything required to update our Census_Tract polygons.
Time for a table-join:
Right click your pseudo_Census_Tracts layer in the table of contents->Joins and Relates->Join.

Again you are presented with the 'Join Data' dialog box, where we need to change the join type back to 'Join attributes from a table'

Then set the join options to:
- OBJECTID
- pseudo_Census_Tracts_with_Neighborhood
- ORIG_FID
- Check Keep all Records

Select OK and open the attribute table for the original pseudo_Census_Tracts polygons layer.

At this point, we just need to perform a field calculation on the original polygons to copy the neccessary data from the joined columns in pseudo_Census_Tracts_with_Neighborhood. Find the column from the orignal pseudo_Census_Tracts table that you wish to calculate the value on. Right click on the column and select Field Calculator.

In the Field Calculator dialog box, select the name of the Joined field that contains the data you wish to copy into the column of your pseudo_Census_Tracts polygon feature layer. *This column should be prefixed with the name of the joined table containing the actual data (here I've chosen pseudo_Census_Tracts_with_Neighborhood.Neighborhood_Name_1 as my data column)

After the field calcultion is comeplete, you can Remove all joins and use the new data within your polygon feature layer. Here I've symbolized the Census_Tracts using the new Neighborhood_Name.
